Copyright (c) 1993 Association of Mizar Users
environ vocabulary FUNCT_1, RELAT_1, BOOLE, PARTFUN1, FRAENKEL, TARSKI, SUPINF_1, ARYTM_3, ORDINAL2, RLVECT_1, RLSUB_1, RLSUB_2, MCART_1, RLVECT_3, GRCAT_1, UNIALG_1, ABSVALUE, FUNCOP_1, ARYTM_1, SEQ_2, NORMSP_1, HAHNBAN, ARYTM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, XREAL_0, STRUCT_0, REAL_1, ABSVALUE, RELAT_1, FUNCT_1, FUNCT_2, RLVECT_1, RLSUB_1, RLSUB_2, NORMSP_1, PARTFUN1, FUNCOP_1, RFUNCT_3, RLVECT_3, FRAENKEL, DOMAIN_1, SUPINF_1; constructors REAL_1, ABSVALUE, RLSUB_2, NORMSP_1, RFUNCT_3, RLVECT_2, RLVECT_3, DOMAIN_1, MEASURE5, MEMBERED, XBOOLE_0; clusters SUBSET_1, FRAENKEL, RLVECT_1, RLSUB_1, SUPINF_1, RELSET_1, STRUCT_0, XREAL_0, PARTFUN1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; definitions TARSKI, RLSUB_1, FRAENKEL, SUPINF_1, RLSUB_2, RLVECT_1, ORDINAL1, XBOOLE_0; theorems PARTFUN1, RLSUB_1, FUNCT_2, ZFMISC_1, FUNCOP_1, REAL_1, AXIOMS, TREES_2, TARSKI, FUNCT_1, FRAENKEL, GRFUNC_1, RLVECT_1, RLSUB_2, RLVECT_4, SUPINF_1, MCART_1, RLVECT_3, ABSVALUE, NORMSP_1, RELAT_1, RELSET_1, XREAL_0, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1; schemes ORDERS_2, DOMAIN_1, FUNCT_2; begin canceled; theorem Th2: for X being set, f,g being Function st X c= dom f & f c= g holds f|X = g|X proof let X be set, f,g be Function such that A1: X c= dom f; assume f c= g; hence f|X = g|(dom f)|X by GRFUNC_1:64 .= g|((dom f) /\ X) by RELAT_1:100 .= g|X by A1,XBOOLE_1:28; end; theorem Th3: for A being non empty set, b being set st A <> {b} ex a being Element of A st a <> b proof let A be non empty set, b be set such that A1: A <> {b}; assume A2: for a being Element of A holds a = b; now let a be set; thus a in A implies a = b by A2; assume A3: a = b; consider a0 being Element of A; a0 = b by A2; hence a in A by A3; end; hence contradiction by A1,TARSKI:def 1; end; theorem Th4: for X,Y being set, B being non empty Subset of PFuncs(X,Y) holds B is non empty functional set proof let X,Y be set, B be non empty Subset of PFuncs(X,Y); B is functional proof let x be set; assume x in B; hence x is Function by PARTFUN1:120; end; hence thesis; end; theorem Th5: for B being non empty functional set, f being Function st f = union B holds dom f = union { dom g where g is Element of B: not contradiction } & rng f = union { rng g where g is Element of B: not contradiction } proof let B be non empty functional set, f be Function such that A1: f = union B; set X = { dom g where g is Element of B: not contradiction }; now let x be set; hereby assume x in dom f; then [x,f.x] in f by FUNCT_1:8; then consider g being set such that A2: [x,f.x] in g & g in B by A1,TARSKI:def 4; reconsider g as Function by A2,FRAENKEL:def 1; take Z = dom g; thus x in Z & Z in X by A2,FUNCT_1:8; end; given Z being set such that A3: x in Z & Z in X; consider g being Element of B such that A4: Z = dom g by A3; [x,g.x] in g by A3,A4,FUNCT_1:8; then [x,g.x] in f by A1,TARSKI:def 4; hence x in dom f by FUNCT_1:8; end; hence dom f = union X by TARSKI:def 4; set X = { rng g where g is Element of B: not contradiction }; now let y be set; hereby assume y in rng f; then consider x being set such that A5: x in dom f & y = f.x by FUNCT_1:def 5; [x,y] in f by A5,FUNCT_1:8; then consider g being set such that A6: [x,y] in g & g in B by A1,TARSKI:def 4; reconsider g as Function by A6,FRAENKEL:def 1; take Z = rng g; x in dom g & y = g.x by A6,FUNCT_1:8; hence y in Z & Z in X by A6,FUNCT_1:def 5; end; given Z being set such that A7: y in Z & Z in X; consider g being Element of B such that A8: Z = rng g by A7; consider x being set such that A9: x in dom g & y = g.x by A7,A8,FUNCT_1:def 5; [x,y] in g by A9,FUNCT_1:8; then [x,y] in f by A1,TARSKI:def 4; hence y in rng f by RELAT_1:20; end; hence rng f = union X by TARSKI:def 4; end; theorem for A being non empty Subset of ExtREAL st for r being R_eal st r in A holds r <=' -infty holds A = {-infty} proof let A be non empty Subset of ExtREAL such that A1: for r being R_eal st r in A holds r <=' -infty; assume A <> {-infty}; then consider a being Element of A such that A2: a <> -infty by Th3; a <=' -infty by A1; hence contradiction by A2,SUPINF_1:23; end; theorem for A being non empty Subset of ExtREAL st for r being R_eal st r in A holds +infty <=' r holds A = {+infty} proof let A be non empty Subset of ExtREAL such that A1: for r being R_eal st r in A holds +infty <=' r; assume A <> {+infty}; then consider a being Element of A such that A2: a <> +infty by Th3; +infty <=' a by A1; hence contradiction by A2,SUPINF_1:24; end; theorem Th8: for A being non empty Subset of ExtREAL, r being R_eal st r <' sup A ex s being R_eal st s in A & r <' s proof let A be non empty Subset of ExtREAL, r be R_eal; assume A1: r <' sup A; assume A2: for s being R_eal st s in A holds not r <' s; r is majorant of A proof let x be R_eal; assume x in A; hence x <=' r by A2; end; hence contradiction by A1,SUPINF_1:def 16; end; theorem Th9: for A being non empty Subset of ExtREAL, r being R_eal st inf A <' r ex s being R_eal st s in A & s <' r proof let A be non empty Subset of ExtREAL, r be R_eal; assume A1: inf A <' r; assume A2: for s being R_eal st s in A holds not s <' r; r is minorant of A proof let x be R_eal; assume x in A; hence r <=' x by A2; end; hence contradiction by A1,SUPINF_1:def 17; end; theorem Th10: for A,B being non empty Subset of ExtREAL st for r,s being R_eal st r in A & s in B holds r <=' s holds sup A <=' inf B proof let A,B be non empty Subset of ExtREAL; assume that A1: for r,s being R_eal st r in A & s in B holds r <=' s; assume not sup A <=' inf B; then consider s1 being R_eal such that A2: s1 in A & inf B <' s1 by Th8; consider s2 being R_eal such that A3: s2 in B & s2 <' s1 by A2,Th9; thus contradiction by A1,A2,A3; end; canceled; theorem Th12: for x,y being R_eal, p,q being real number st x = p & y = q holds (p <= q iff x <=' y) proof let x,y be R_eal; let p,q be real number; assume A1:x = p & y = q; reconsider p, q as Real by XREAL_0:def 1; A2:x = p & y = q by A1; x <=' y implies p <= q proof assume x <=' y; then consider p1,q1 being Real such that A3:p1 = x & q1 = y & p1 <= q1 by A2,SUPINF_1:16; thus thesis by A1,A3; end; hence thesis by A2,SUPINF_1:16; end; begin :: Chains of Functions definition let A be non empty set; cluster c=-linear non empty Subset of A; existence proof consider a being Element of A; reconsider B = { a } as non empty Subset of A; take B; B is c=-linear proof let x,y be set; assume x in B & y in B; then x = a & y = a by TARSKI:def 1; hence thesis; end; hence thesis; end; end; theorem Th13: for X,Y being set for B being c=-linear Subset of PFuncs(X,Y) holds union B in PFuncs(X,Y) proof let X,Y be set; let B be c=-linear Subset of PFuncs(X,Y); for x be set st x in B holds x is Function by PARTFUN1:120; then reconsider f = union B as Function by TREES_2:36; per cases; suppose B <> {}; then reconsider D = B as non empty functional set by Th4; A1: now let x be set; assume x in { dom g where g is Element of D: not contradiction }; then consider g being Element of D such that A2: x = dom g; g in PFuncs(X,Y) by TARSKI:def 3; then ex f being Function st g = f & dom f c= X & rng f c= Y by PARTFUN1:def 5; hence x c= X by A2; end; A3: now let x be set; assume x in { rng g where g is Element of D: not contradiction }; then consider g being Element of D such that A4: x = rng g; g in PFuncs(X,Y) by TARSKI:def 3; then ex f being Function st g = f & dom f c= X & rng f c= Y by PARTFUN1:def 5; hence x c= Y by A4; end; dom f = union { dom g where g is Element of D: not contradiction } & rng f = union { rng g where g is Element of D: not contradiction } by Th5; then dom f c= X & rng f c= Y by A1,A3,ZFMISC_1:94; hence union B in PFuncs(X,Y) by PARTFUN1:def 5; suppose A5: B = {}; {} is PartFunc of X, Y by PARTFUN1:56; hence thesis by A5,PARTFUN1:119,ZFMISC_1:2; end; begin :: Some Facts on Real Vector Spaces reserve V for RealLinearSpace; theorem Th14: for W1,W2 being Subspace of V holds the carrier of W1 c= the carrier of W1 + W2 proof let W1,W2 be Subspace of V; let x be set; assume A1: x in the carrier of W1; the carrier of W1 c= the carrier of V by RLSUB_1:def 2; then reconsider w = x as VECTOR of V by A1; A2: w + 0.V = w by RLVECT_1:10; x in W1 & 0.V in W2 by A1,RLSUB_1:25,RLVECT_1:def 1; then x in {v + u where v,u is VECTOR of V : v in W1 & u in W2} by A2; hence x in the carrier of W1 + W2 by RLSUB_2:def 1; end; theorem Th15: for W1,W2 being Subspace of V st V is_the_direct_sum_of W1,W2 for v,v1,v2 being VECTOR of V st v1 in W1 & v2 in W2 & v = v1 + v2 holds v |-- (W1,W2) = [v1,v2] proof let W1,W2 be Subspace of V; assume A1: V is_the_direct_sum_of W1,W2; let v,v1,v2 be VECTOR of V; [v1,v2]`1 = v1 & [v1,v2]`2 = v2 by MCART_1:7; hence thesis by A1,RLSUB_2:def 6; end; theorem Th16: for W1,W2 being Subspace of V st V is_the_direct_sum_of W1,W2 for v,v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2] holds v = v1 + v2 proof let W1,W2 be Subspace of V such that A1: V is_the_direct_sum_of W1,W2; let v,v1,v2 be VECTOR of V; assume v |-- (W1,W2) = [v1,v2]; then (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2 by MCART_1:7; hence v = v1 + v2 by A1,RLSUB_2:def 6; end; theorem Th17: for W1,W2 being Subspace of V st V is_the_direct_sum_of W1,W2 for v,v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2] holds v1 in W1 & v2 in W2 proof let W1,W2 be Subspace of V such that A1: V is_the_direct_sum_of W1,W2; let v,v1,v2 be VECTOR of V; assume v |-- (W1,W2) = [v1,v2]; then (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2 by MCART_1:7; hence thesis by A1,RLSUB_2:def 6; end; theorem Th18: for W1,W2 being Subspace of V st V is_the_direct_sum_of W1,W2 for v,v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2] holds v |-- (W2,W1) = [v2,v1] proof let W1,W2 be Subspace of V; assume A1: V is_the_direct_sum_of W1,W2; then A2: V is_the_direct_sum_of W2,W1 by RLSUB_2:46; let v,v1,v2 be VECTOR of V; assume v |-- (W1,W2) = [v1,v2]; then A3: (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2 by MCART_1:7; then A4: v1 in W1 & v2 in W2 by A1,RLSUB_2:def 6; v = v2 + v1 by A1,A3,RLSUB_2:def 6; hence v |-- (W2,W1) = [v2,v1] by A2,A4,Th15; end; theorem Th19: for W1,W2 being Subspace of V st V is_the_direct_sum_of W1,W2 for v being VECTOR of V st v in W1 holds v |-- (W1,W2) = [v,0.V] proof let W1,W2 be Subspace of V such that A1: V is_the_direct_sum_of W1,W2; let v be VECTOR of V such that A2: v in W1; A3: 0.V in W2 by RLSUB_1:25; v + 0.V = v by RLVECT_1:10; hence v |-- (W1,W2) = [v,0.V] by A1,A2,A3,Th15; end; theorem Th20: for W1,W2 being Subspace of V st V is_the_direct_sum_of W1,W2 for v being VECTOR of V st v in W2 holds v |-- (W1,W2) = [0.V,v] proof let W1,W2 be Subspace of V; assume V is_the_direct_sum_of W1,W2; then A1: V is_the_direct_sum_of W2,W1 by RLSUB_2:46; let v be VECTOR of V; assume v in W2; then v |-- (W2,W1) = [v,0.V] by A1,Th19; hence v |-- (W1,W2) = [0.V,v] by A1,Th18; end; theorem Th21: for V1 being Subspace of V, W1 being Subspace of V1, v being VECTOR of V st v in W1 holds v is VECTOR of V1 proof let V1 be Subspace of V, W1 be Subspace of V1, v be VECTOR of V such that A1: v in W1; A2: the carrier of W1 c= the carrier of V1 by RLSUB_1:def 2; v in the carrier of W1 by A1,RLVECT_1:def 1; hence v is VECTOR of V1 by A2; end; theorem Th22: for V1,V2,W being Subspace of V, W1,W2 being Subspace of W st W1 = V1 & W2 = V2 holds W1 + W2 = V1 + V2 proof let V1,V2,W be Subspace of V, W1,W2 be Subspace of W such that A1: W1 = V1 & W2 = V2; reconsider W3 = W1 + W2 as Subspace of V by RLSUB_1:35; now let v be VECTOR of V; hereby assume A2: v in W3; then reconsider w = v as VECTOR of W by Th21; consider w1,w2 being VECTOR of W such that A3: w1 in W1 & w2 in W2 and A4: w = w1 + w2 by A2,RLSUB_2:5; reconsider v1 = w1, v2 = w2 as VECTOR of V by RLSUB_1:18; v = v1 + v2 by A4,RLSUB_1:21; hence v in V1 + V2 by A1,A3,RLSUB_2:5; end; A5: the carrier of W1 c= the carrier of W & the carrier of W2 c= the carrier of W by RLSUB_1:def 2; assume v in V1 + V2; then consider v1,v2 being VECTOR of V such that A6: v1 in V1 & v2 in V2 and A7: v = v1 + v2 by RLSUB_2:5; v1 in the carrier of W1 & v2 in the carrier of W2 by A1,A6,RLVECT_1:def 1; then reconsider w1 = v1, w2 = v2 as VECTOR of W by A5; v = w1 + w2 by A7,RLSUB_1:21; hence v in W3 by A1,A6,RLSUB_2:5; end; hence W1 + W2 = V1 + V2 by RLSUB_1:39; end; theorem Th23: for W being Subspace of V for v being VECTOR of V, w being VECTOR of W st v = w holds Lin{w} = Lin{v} proof let W be Subspace of V; let v be VECTOR of V, w be VECTOR of W such that A1: v = w; reconsider W1 = Lin{w} as Subspace of V by RLSUB_1:35; now let u be VECTOR of V; hereby assume u in W1; then consider a being Real such that A2: u = a * w by RLVECT_4:11; u = a * v by A1,A2,RLSUB_1:22; hence u in Lin{v} by RLVECT_4:11; end; assume u in Lin{v}; then consider a being Real such that A3: u = a * v by RLVECT_4:11; u = a * w by A1,A3,RLSUB_1:22; hence u in W1 by RLVECT_4:11; end; hence Lin{w} = Lin{v} by RLSUB_1:39; end; theorem Th24: for v being VECTOR of V, X being Subspace of V st not v in X for y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & W = X holds X + Lin{v} is_the_direct_sum_of W,Lin{y} proof let v be VECTOR of V, X be Subspace of V such that A1: not v in X; let y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v}; assume A2: v = y & W = X; then Lin{v} = Lin{y} by Th23; hence the RLSStruct of X + Lin{v} = W + Lin{y} by A2,Th22; assume W /\ Lin{y} <> (0).(X + Lin{v}); then consider z being VECTOR of X + Lin{v} such that A3: not(z in W /\ Lin{y} iff z in (0).(X + Lin{v})) by RLSUB_1:39; per cases by A3; suppose that A4: z in W /\ Lin{y} and A5: not z in (0).(X + Lin{v}); A6: z in W by A4,RLSUB_2:7; z in Lin{y} by A4,RLSUB_2:7; then consider a being Real such that A7: z = a * y by RLVECT_4:11; now per cases; suppose a = 0; then z = 0.(X + Lin{v}) by A7,RLVECT_1:23; hence contradiction by A5,RLSUB_1:25; suppose A8: a <> 0; y = 1*y by RLVECT_1:def 9 .= a"*a*y by A8,XCMPLX_0:def 7 .= a"*(a*y) by RLVECT_1:def 9; hence contradiction by A1,A2,A6,A7,RLSUB_1:29; end; hence contradiction; suppose that A9: not z in W /\ Lin{y} and A10: z in (0).(X + Lin{v}); z = 0.(X + Lin{v}) by A10,RLVECT_3:36; hence contradiction by A9,RLSUB_1:25; end; theorem Th25: for v being VECTOR of V, X being Subspace of V, y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & X = W & not v in X holds y |-- (W,Lin{y}) = [0.W,y] proof let v be VECTOR of V, X be Subspace of V, y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v}; A1: y in Lin{y} by RLVECT_4:12; assume v = y & X = W & not v in X; then X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th24; then y |-- (W,Lin{y}) = [0.(X + Lin{v}),y] by A1,Th20; hence y |-- (W,Lin{y}) = [0.W,y] by RLSUB_1:19; end; theorem Th26: for v being VECTOR of V, X being Subspace of V, y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & X = W & not v in X for w being VECTOR of X + Lin{v} st w in X holds w |-- (W,Lin{y}) = [w,0.V] proof let v be VECTOR of V, X be Subspace of V, y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v} such that A1: v = y & X = W & not v in X; let w be VECTOR of X + Lin{v} such that A2: w in X; X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,Th24; then w |-- (W,Lin{y}) = [w,0.(X + Lin{v})] by A1,A2,Th19; hence w |-- (W,Lin{y}) = [w,0.V] by RLSUB_1:19; end; theorem Th27: for v being VECTOR of V, W1,W2 being Subspace of V ex v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2] proof let v be VECTOR of V, W1,W2 be Subspace of V; take (v |-- (W1,W2))`1,(v |-- (W1,W2))`2; thus v |-- (W1,W2) = [(v |-- (W1,W2))`1,v |-- (W1,W2)`2] by MCART_1:23; end; theorem Th28: for v being VECTOR of V, X being Subspace of V, y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & X = W & not v in X for w being VECTOR of X + Lin{v} ex x being VECTOR of X, r being Real st w |-- (W,Lin{y}) = [x,r*v] proof let v be VECTOR of V, X be Subspace of V, y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v} such that A1: v = y & X = W & not v in X; let w be VECTOR of X + Lin{v}; A2: X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,Th24; consider v1,v2 being VECTOR of X + Lin{v} such that A3: w |-- (W,Lin{y}) = [v1,v2] by Th27; v1 in W by A2,A3,Th17; then reconsider x = v1 as VECTOR of X by A1,RLVECT_1:def 1; v2 in Lin{y} by A2,A3,Th17; then consider r being Real such that A4: v2 = r * y by RLVECT_4:11; take x,r; thus w |-- (W,Lin{y}) = [x,r*v] by A1,A3,A4,RLSUB_1:22; end; theorem Th29: for v being VECTOR of V, X being Subspace of V, y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & X = W & not v in X for w1,w2 being VECTOR of X + Lin{v}, x1,x2 being VECTOR of X, r1,r2 being Real st w1 |-- (W,Lin{y}) = [x1,r1*v] & w2 |-- (W,Lin{y}) = [x2,r2*v] holds w1 + w2 |-- (W,Lin{y}) = [x1 + x2, (r1+r2)*v] proof let v be VECTOR of V, X be Subspace of V, y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v}; assume A1: v = y & X = W & not v in X; then A2: X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th24; let w1,w2 be VECTOR of X + Lin{v}, x1,x2 be VECTOR of X, r1,r2 be Real such that A3: w1 |-- (W,Lin{y}) = [x1,r1*v] & w2 |-- (W,Lin{y}) = [x2,r2*v]; reconsider z1 = x1, z2 = x2 as VECTOR of V by RLSUB_1:18; reconsider y1 = x1, y2 = x2 as VECTOR of X + Lin{v} by A1,RLSUB_1:18; A4: r1*v = r1*y & r2*v = r2*y by A1,RLSUB_1:22; then y1 in W & y2 in W by A2,A3,Th17; then A5: y1 + y2 in W by RLSUB_1:28; (r1+r2)*v = (r1+r2)*y by A1,RLSUB_1:22; then A6: (r1+r2)*v in Lin{y} by RLVECT_4:11; w1 = y1 + r1*y & w2 = y2 + r2*y by A2,A3,A4,Th16; then A7: w1 + w2 = y1 + r1*y + y2 + r2*y by RLVECT_1:def 6 .= y1 + y2 + r1*y + r2*y by RLVECT_1:def 6 .= y1 + y2 + (r1*y + r2*y) by RLVECT_1:def 6 .= y1 + y2 + (r1+r2)*y by RLVECT_1:def 9; A8: (r1+r2)*y = (r1+r2)*v by A1,RLSUB_1:22; y1 + y2 = z1 + z2 by RLSUB_1:21 .= x1 + x2 by RLSUB_1:21; hence w1 + w2 |-- (W,Lin{y}) = [x1 + x2, (r1+r2)*v] by A2,A5,A6,A7,A8,Th15; end; theorem Th30: for v being VECTOR of V, X being Subspace of V, y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & X = W & not v in X for w being VECTOR of X + Lin{v}, x being VECTOR of X, t,r being Real st w |-- (W,Lin{y}) = [x,r*v] holds t*w |-- (W,Lin{y}) = [t*x, t*r*v] proof let v be VECTOR of V, X be Subspace of V, y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v}; assume A1: v = y & X = W & not v in X; then A2: X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th24; let w be VECTOR of X + Lin{v}, x be VECTOR of X, t,r be Real such that A3: w |-- (W,Lin{y}) = [x,r*v]; reconsider z = x as VECTOR of X + Lin{v} by A1,RLSUB_1:18; reconsider u = x as VECTOR of V by RLSUB_1:18; A4: t*z = t*u by RLSUB_1:22 .= t*x by RLSUB_1:22; A5: t*r*y = t*r*v by A1,RLSUB_1:22; A6: t*z in W by A1,A4,RLVECT_1:def 1; A7: t*r*y in Lin{y} by RLVECT_4:11; r*y = r*v by A1,RLSUB_1:22; then t*w = t*(z + r*y) by A2,A3,Th16 .= t*z + t*(r*y) by RLVECT_1:def 9 .= t*z + t*r*y by RLVECT_1:def 9; hence t*w |-- (W,Lin{y}) = [t*x, t*r*v] by A2,A4,A5,A6,A7,Th15; end; begin :: Functionals in Real Linear Space definition let V be RLSStruct; canceled 2; mode Functional of V is Function of the carrier of V,REAL; end; definition let V; let IT be Functional of V; attr IT is subadditive means :Def3: for x,y being VECTOR of V holds IT.(x+y) <= IT.x+IT.y; attr IT is additive means :Def4: for x,y being VECTOR of V holds IT.(x+y) = IT.x+IT.y; attr IT is homogeneous means :Def5: for x being VECTOR of V, r being Real holds IT.(r*x) = r*IT.x; attr IT is positively_homogeneous means :Def6: for x being VECTOR of V, r being Real st r > 0 holds IT.(r*x) = r*IT.x; attr IT is semi-homogeneous means :Def7: for x being VECTOR of V, r being Real st r >= 0 holds IT.(r*x) = r*IT.x; attr IT is absolutely_homogeneous means :Def8: for x being VECTOR of V, r being Real holds IT.(r*x) = abs(r)*IT.x; attr IT is 0-preserving means :Def9: IT.(0.V) = 0; end; definition let V; cluster additive -> subadditive Functional of V; coherence proof let f be Functional of V; assume A1: f is additive; let x,y be VECTOR of V; thus f.(x+y) <= f.x+f.y by A1,Def4; end; cluster homogeneous -> positively_homogeneous Functional of V; coherence proof let f be Functional of V; assume A2: f is homogeneous; let x be VECTOR of V, r be Real; thus thesis by A2,Def5; end; cluster semi-homogeneous -> positively_homogeneous Functional of V; coherence proof let f be Functional of V; assume A3: f is semi-homogeneous; let x be VECTOR of V, r be Real; assume r > 0; hence thesis by A3,Def7; end; cluster semi-homogeneous -> 0-preserving Functional of V; coherence proof let f be Functional of V; assume A4: f is semi-homogeneous; thus f.(0.V) = f.(0*0.V) by RLVECT_1:23 .= 0*f.(0.V) by A4,Def7 .= 0; end; cluster absolutely_homogeneous -> semi-homogeneous Functional of V; coherence proof let f be Functional of V; assume A5: f is absolutely_homogeneous; let x be VECTOR of V, r be Real; assume r >= 0; then abs r = r by ABSVALUE:def 1; hence thesis by A5,Def8; end; cluster 0-preserving positively_homogeneous -> semi-homogeneous Functional of V; coherence proof let f be Functional of V such that A6: f is 0-preserving and A7: f is positively_homogeneous; let x be VECTOR of V, r be Real such that A8: r >= 0; per cases by A8; suppose A9: r = 0; hence f.(r*x) = f.(0.V) by RLVECT_1:23 .= r*f.x by A6,A9,Def9; suppose r > 0; hence f.(r*x) = r*f.x by A7,Def6; end; end; definition let V; cluster additive absolutely_homogeneous homogeneous Functional of V; existence proof reconsider f = (the carrier of V) --> 0 as Functional of V by FUNCOP_1:58; take f; hereby let x,y be VECTOR of V; thus f.(x+y) = 0+0 by FUNCOP_1:13 .= f.x+0 by FUNCOP_1:13 .= f.x+f.y by FUNCOP_1:13; end; hereby let x be VECTOR of V, r be Real; thus f.(r*x) = abs(r)*0 by FUNCOP_1:13 .= abs(r)*f.x by FUNCOP_1:13; end; let x be VECTOR of V, r be Real; thus f.(r*x) = r*0 by FUNCOP_1:13 .= r*f.x by FUNCOP_1:13; end; end; definition let V; mode Banach-Functional of V is subadditive positively_homogeneous Functional of V; mode linear-Functional of V is additive homogeneous Functional of V; end; theorem Th31: for L being homogeneous Functional of V, v being VECTOR of V holds L.(-v) = - L.v proof let L be homogeneous Functional of V, v be VECTOR of V; thus L.(-v) = L.((-1)*v) by RLVECT_1:29 .= (-1)*L.v by Def5 .= - L.v by XCMPLX_1:180; end; theorem Th32: for L being linear-Functional of V, v1,v2 being VECTOR of V holds L.(v1 - v2) = L.v1 - L.v2 proof let L be linear-Functional of V, v1,v2 be VECTOR of V; thus L.(v1 - v2) = L.(v1 + -v2) by RLVECT_1:def 11 .= L.(v1) + L.(-v2) by Def4 .= L.v1 + - L.v2 by Th31 .= L.v1 - L.v2 by XCMPLX_0:def 8; end; theorem Th33: for L being additive Functional of V holds L.(0.V) = 0 proof let L be additive Functional of V; L.(0.V) + 0 = L.(0.V + 0.V) by RLVECT_1:10 .= L.(0.V) + L.(0.V) by Def4; hence L.(0.V) = 0 by XCMPLX_1:2; end; theorem Th34: for X being Subspace of V, fi being linear-Functional of X, v being VECTOR of V, y being VECTOR of X + Lin{v} st v = y & not v in X for r being Real ex psi being linear-Functional of X + Lin{v} st psi|the carrier of X=fi & psi.y = r proof let X be Subspace of V, fi be linear-Functional of X, v be VECTOR of V, y be VECTOR of X + Lin{v} such that A1: v = y and A2: not v in X; let r be Real; reconsider W1 = X as Subspace of X + Lin{v} by RLSUB_2:11; defpred P[VECTOR of X + Lin{v},Real] means for x being VECTOR of X, s being Real st ($1 |-- (W1,Lin{y}))`1 = x & ($1 |-- (W1,Lin{y}))`2 = s*v holds $2 = fi.x + s*r; A3: now let e be Element of X + Lin{v}; consider x being VECTOR of X, s being Real such that A4: e |-- (W1,Lin{y}) = [x,s*v] by A1,A2,Th28; take u = fi.x + s*r; thus P[e,u] proof let x' be VECTOR of X, t be Real such that A5: (e |-- (W1,Lin{y}))`1 = x' & (e |-- (W1,Lin{y}))`2 = t*v; A6: v <> 0.V by A2,RLSUB_1:25; s*v = t * v by A4,A5,MCART_1:7; then t = s by A6,RLVECT_1:51; hence u = fi.x' + t*r by A4,A5,MCART_1:7; end; end; consider f being Function of the carrier of X + Lin{v},REAL such that A7: for e being VECTOR of X + Lin{v} holds P[e,f.e] from FuncExD(A3); A8: dom fi = the carrier of X by FUNCT_2:def 1; A9: now let a be set; assume a in dom fi; then reconsider x = a as VECTOR of X by FUNCT_2:def 1; the carrier of X c= the carrier of X + Lin{v} by Th14; then reconsider v1 = x as VECTOR of X + Lin{v} by TARSKI:def 3; v1 in X by RLVECT_1:def 1; then (v1 |-- (W1,Lin{y})) = [v1,0.V] by A1,A2,Th26 .= [v1,0*v] by RLVECT_1:23; then A10: (v1 |-- (W1,Lin{y}))`1 = x & (v1 |-- (W1,Lin{y}))`2 = 0*v by MCART_1:7; thus fi.a = fi.x + 0*r .= f.a by A7,A10; end; reconsider f as Functional of X + Lin{v}; A11: f is additive proof let v1,v2 be VECTOR of X + Lin{v}; consider x1 being VECTOR of X, s1 being Real such that A12: v1 |-- (W1,Lin{y}) = [x1,s1*v] by A1,A2,Th28; consider x2 being VECTOR of X, s2 being Real such that A13: v2 |-- (W1,Lin{y}) = [x2,s2*v] by A1,A2,Th28; A14: (v1 |-- (W1,Lin{y}))`1 = x1 & (v1 |-- (W1,Lin{y}))`2 = s1*v by A12,MCART_1:7; A15: (v2 |-- (W1,Lin{y}))`1 = x2 & (v2 |-- (W1,Lin{y}))`2 = s2*v by A13,MCART_1:7; v1 + v2 |-- (W1,Lin{y}) = [x1 +x2,(s1+s2)*v] by A1,A2,A12,A13,Th29; then (v1 + v2 |-- (W1,Lin{y}))`1 = x1 + x2 & (v1 + v2 |-- (W1,Lin{y}))`2 = (s1+ s2)*v by MCART_1:7; hence f.(v1+v2) = fi.(x1 + x2) + (s1 + s2)*r by A7 .= fi.(x1 + x2) + (s1*r + s2*r) by XCMPLX_1:8 .= fi.(x1) + fi.(x2) + (s1*r + s2*r) by Def4 .= fi.(x1) + fi.(x2) + s1*r + s2*r by XCMPLX_1:1 .= fi.(x1) + s1*r + fi.(x2) + s2*r by XCMPLX_1:1 .= fi.(x1) + s1*r + (fi.(x2) + s2*r) by XCMPLX_1:1 .= f.v1 + (fi.(x2) + s2*r) by A7,A14 .= f.v1+f.v2 by A7,A15; end; f is homogeneous proof let v0 be VECTOR of X + Lin{v}, t be Real; consider x0 being VECTOR of X, s0 being Real such that A16: v0 |-- (W1,Lin{y}) = [x0,s0*v] by A1,A2,Th28; A17: (v0 |-- (W1,Lin{y}))`1 = x0 & (v0 |-- (W1,Lin{y}))`2 = s0*v by A16,MCART_1:7; t*v0 |-- (W1,Lin{y}) = [t*x0,t*s0*v] by A1,A2,A16,Th30; then (t*v0 |-- (W1,Lin{y}))`1 = t*x0 & (t*v0 |-- (W1,Lin{y}))`2 = t*s0*v by MCART_1:7; hence f.(t*v0) = fi.(t*x0) + t*s0*r by A7 .= t*(fi.x0) + t*s0*r by Def5 .= t*(fi.x0) + t*(s0*r) by XCMPLX_1:4 .= t*(fi.x0 + s0*r) by XCMPLX_1:8 .= t*f.v0 by A7,A17; end; then reconsider f as linear-Functional of X + Lin{v} by A11; take f; dom f = the carrier of X + Lin{v} by FUNCT_2:def 1; then dom fi c= dom f by A8,Th14; then dom fi = dom f /\ the carrier of X by A8,XBOOLE_1:28; hence f|the carrier of X=fi by A9,FUNCT_1:68; A18: y |-- (W1,Lin{y}) = [0.W1,y] by A1,A2,Th25; then A19: (y |-- (W1,Lin{y}))`1 = 0.X by MCART_1:7; (y |-- (W1,Lin{y}))`2 = y by A18,MCART_1:7 .= 1*v by A1,RLVECT_1:def 9; hence f.y = fi.(0.X) + 1*r by A7,A19 .= 0 + 1*r by Th33 .= r; end; begin :: Hahn Banach Theorem Lm1: for X being Subspace of V, v being VECTOR of V st not v in the carrier of X for q being Banach-Functional of V, fi being linear-Functional of X st for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v ex psi being linear-Functional of X + Lin{v} st psi|the carrier of X=fi & for x being VECTOR of X + Lin{v}, v being VECTOR of V st x = v holds psi.x <= q.v proof let X be Subspace of V, v be VECTOR of V such that A1: not v in the carrier of X; let q be Banach-Functional of V, fi be linear-Functional of X such that A2: for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v; A3: the carrier of Lin{v} c= the carrier of Lin{v} + X by Th14; v in Lin{v} by RLVECT_4:12; then v in the carrier of Lin{v} by RLVECT_1:def 1; then reconsider x0 = v as VECTOR of X + Lin{v} by A3,RLSUB_2:9; consider x1 being VECTOR of X; reconsider v1 = x1 as VECTOR of V by RLSUB_1:18; set A = { fi.x - q.(y - v) where x is VECTOR of X, y is VECTOR of V : x = y }, B = { fi.x + q.(v - y) where x is VECTOR of X, y is VECTOR of V : x = y }; A4: A c= REAL proof let a be set; assume a in A; then ex x being VECTOR of X, y being VECTOR of V st a = fi.x - q.(y - v) & x = y; hence a in REAL; end; A5: B c= REAL proof let b be set; assume b in B; then ex x being VECTOR of X, y being VECTOR of V st b = fi.x + q.(v - y) & x = y; hence b in REAL; end; A6: now let x1,x2 be VECTOR of X, y1,y2 be VECTOR of V; assume x1 = y1 & x2 = y2; then x1 - x2 = y1 - y2 by RLSUB_1:24; then fi.(x1 - x2) <= q.(y1 - y2) by A2; then A7: fi.x1 - fi.x2 <= q.(y1 - y2) by Th32; A8: y1 + -v = y1 - v & v + -y2 = v - y2 by RLVECT_1:def 11; y1 - y2 = y1 + -y2 by RLVECT_1:def 11 .= y1 + 0.V + -y2 by RLVECT_1:10 .= y1 + (-v + v) + -y2 by RLVECT_1:16 .= y1 - v + v + -y2 by A8,RLVECT_1:def 6 .= y1 - v + (v - y2) by A8,RLVECT_1:def 6; then q.(y1 - y2) <= q.(y1 - v) + q.(v - y2) by Def3; then fi.x1 - fi.x2 <= q.(v - y2) + q.(y1 - v) by A7,AXIOMS:22; then fi.x1 <= fi.x2 + (q.(v - y2) + q.(y1 - v)) by REAL_1:86; then fi.x1 <= fi.x2 + q.(v - y2) + q.(y1 - v) by XCMPLX_1:1; hence fi.x1 - q.(y1 - v) <= fi.x2 + q.(v - y2) by REAL_1:86; end; A9: now let a,b be R_eal; assume a in A; then consider x1 being VECTOR of X, y1 being VECTOR of V such that A10: a = fi.x1 - q.(y1 - v) & x1 = y1; assume b in B; then consider x2 being VECTOR of X, y2 being VECTOR of V such that A11: b = fi.x2 + q.(v - y2) & x2 = y2; fi.x1 - q.(y1 - v) <= fi.x2 + q.(v - y2) by A6,A10,A11; hence a <=' b by A10,A11,SUPINF_1:def 7; end; fi.x1 - q.(v1 - v) in A & fi.x1 + q.(v - v1) in B; then reconsider A, B as non empty Subset of ExtREAL by A4,A5,XBOOLE_1:1; A12: sup A <=' inf B by A9,Th10; A13: A <> {-infty} proof consider x being Element of A; assume A = {-infty}; then x = -infty & x in REAL by A4,TARSKI:def 1,def 3; hence contradiction by SUPINF_1:6; end; A is bounded_above proof consider b being Element of B; reconsider b as Element of ExtREAL; b is majorant of A proof let x be R_eal; thus thesis by A9; end; then reconsider b as majorant of A; take b; thus thesis by A5,TARSKI:def 3; end; then reconsider r = sup A as Real by A13,SUPINF_1:83; not v in X by A1,RLVECT_1:def 1; then consider psi being linear-Functional of X + Lin{v} such that A14: psi|the carrier of X=fi and A15: psi.x0 = r by Th34; take psi; thus psi|the carrier of X=fi by A14; let y be VECTOR of X + Lin{v}, u be VECTOR of V such that A16: y = u; y in X + Lin{v} by RLVECT_1:def 1; then consider y1,y2' being VECTOR of V such that A17: y1 in X & y2' in Lin{v} and A18: y = y1 + y2' by RLSUB_2:5; consider t being Real such that A19: y2' = t * v by A17,RLVECT_4:11; y1 in X + Lin{v} by A17,RLSUB_2:6; then reconsider x = y1 as VECTOR of X + Lin{v} by RLVECT_1:def 1; reconsider x1 = x as VECTOR of X by A17,RLVECT_1:def 1; per cases; suppose t = 0; then y2' = 0.V by A19,RLVECT_1:23; then A20: y = x1 by A18,RLVECT_1:10; then fi.x1 <= q.u by A2,A16; hence thesis by A14,A20,FUNCT_1:72; suppose A21: t > 0; y2' in X + Lin{v} by A17,RLSUB_2:6; then reconsider y2 = y2' as VECTOR of X + Lin{v} by RLVECT_1:def 1; y2 = t*x0 by A19,RLSUB_1:22; then A22: y = x + t*x0 by A18,RLSUB_1:21; A23: t" > 0 by A21,REAL_1:72; A24: (-t)"*x1 = (-t)"*y1 by RLSUB_1:22; then (-t)"*x1 = (-t)"*x by RLSUB_1:22; then A25: fi.((-t)"*x1) = psi.((-t)"*x) by A14,FUNCT_1:72; A26: t*(t"*(q.u)) = t*t"*(q.u) by XCMPLX_1:4 .= 1*(q.u) by A21,XCMPLX_0:def 7 .= q.u; A27: psi.(x + t*x0) = psi.x + psi.(t*x0) by Def4 .= 1*(psi.x + t*psi.x0) by Def5 .= t*t"*(psi.x + t*psi.x0) by A21,XCMPLX_0:def 7 .= t*(t"*(psi.x + t*psi.x0)) by XCMPLX_1:4 .= t*(t"*(psi.x) + t"*(t*psi.x0)) by XCMPLX_1:8 .= t*(t"*(psi.x) + t"*t*psi.x0) by XCMPLX_1:4 .= t*(t"*(psi.x) + 1*psi.x0) by A21,XCMPLX_0:def 7 .= t*(psi.(t"*x) + r) by A15,Def5; reconsider b = psi.((-t)"*x) + q.(v - (-t)"*y1) as R_eal by SUPINF_1:def 6, XBOOLE_0:def 2; psi.((-t)"*x) + q.(v - (-t)"*y1) in B by A24,A25; then inf B <=' b by SUPINF_1:79; then sup A <=' b by A12,SUPINF_1:29; then psi.((-t)"*x) + q.(v - (-t)"*y1) >= r by Th12; then psi.((-t)"*x) >= r - q.(v - (-t)"*y1) by REAL_1:86; then A28: - psi.((-t)"*x) <= - (r - q.(v - (-t)"*y1)) by REAL_1:50; A29: - psi.((-t)"*x) = (-1)*psi.((-t)"*x) by XCMPLX_1:180 .= psi.((-1)*((-t)"*x)) by Def5 .= psi.((-1)*(-t)"*x) by RLVECT_1:def 9 .= psi.((-1)*(-t")*x) by XCMPLX_1:224 .= psi.(1*t"*x) by XCMPLX_1:177 .= psi.(t"*x); v - (-t)"*y1 = v - (-t")*y1 by XCMPLX_1:224 .= v - -t"*y1 by RLVECT_4:6 .= v + - -t"*y1 by RLVECT_1:def 11 .= v + t"*y1 by RLVECT_1:30; then A30: psi.(t"*x) <= q.(v + t"*y1) - r by A28,A29,XCMPLX_1:143; t"*u = t"*y1 + t"*y2' by A16,A18,RLVECT_1:def 9 .= t"*y1 + t"*t*v by A19,RLVECT_1:def 9 .= t"*y1 + 1*v by A21,XCMPLX_0:def 7 .= v + t"*y1 by RLVECT_1:def 9; then psi.(t"*x) + r <= q.(t"*u) by A30,REAL_1:84; then psi.(t"*x) + r <= t"*(q.u) by A23,Def6; hence psi.y <= q.u by A21,A22,A26,A27,AXIOMS:25; suppose t < 0; then A31: -t > 0 by REAL_1:66; A32: -y2' = (-t)*v by A19,RLVECT_4:6; A33: -y2' in Lin{v} by A17,RLSUB_1:30; A34: y = y1 + --y2' by A18,RLVECT_1:30 .= y1 - -y2' by RLVECT_1:def 11; -y2' in X + Lin{v} by A33,RLSUB_2:6; then reconsider y2 = -y2' as VECTOR of X + Lin{v} by RLVECT_1:def 1; y2 = (-t)*x0 by A32,RLSUB_1:22; then A35: y = x - (-t)*x0 by A34,RLSUB_1:24; A36: (-t)" > 0 by A31,REAL_1:72; A37: (-t)"*x1 = (-t)"*y1 by RLSUB_1:22; then (-t)"*x1 = (-t)"*x by RLSUB_1:22; then A38: fi.((-t)"*x1) = psi.((-t)"*x) by A14,FUNCT_1:72; A39: (-t)*((-t)"*(q.u)) = (-t)*(-t)"*(q.u) by XCMPLX_1:4 .= 1*(q.u) by A31,XCMPLX_0:def 7 .= q.u; A40: psi.(x - (-t)*x0) = psi.x - psi.((-t)*x0) by Th32 .= 1*(psi.x - (-t)*psi.x0) by Def5 .= (-t)*(-t)"*(psi.x - (-t)*psi.x0) by A31,XCMPLX_0:def 7 .= (-t)*((-t)"*(psi.x - (-t)*psi.x0)) by XCMPLX_1:4 .= (-t)*((-t)"*(psi.x) - (-t)"*((-t)*psi.x0)) by XCMPLX_1:40 .= (-t)*((-t)"*(psi.x) - (-t)"*(-t)*psi.x0) by XCMPLX_1:4 .= (-t)*((-t)"*(psi.x) - 1*psi.x0) by A31,XCMPLX_0:def 7 .= (-t)*(psi.((-t)"*x) - r) by A15,Def5; reconsider a = psi.((-t)"*x) - q.((-t)"*y1 - v) as R_eal by SUPINF_1:def 6, XBOOLE_0:def 2; psi.((-t)"*x) - q.((-t)"*y1 - v) in A by A37,A38; then a <=' sup A by SUPINF_1:76; then psi.((-t)"*x) - q.((-t)"*y1 - v) <= r by Th12; then A41: psi.((-t)"*x) <= r + q.((-t)"*y1 - v) by REAL_1:86; (-t)"*u = (-t)"*y1 - (-t)"*-y2' by A16,A34,RLVECT_1:48 .= (-t)"*y1 - (-t)"*(-t)*v by A32,RLVECT_1:def 9 .= (-t)"*y1 - 1*v by A31,XCMPLX_0:def 7 .= (-t)"*y1 - v by RLVECT_1:def 9; then psi.((-t)"*x) - r <= q.((-t)"*u) by A41,REAL_1:86; then psi.((-t)"*x) - r <= (-t)"*(q.u) by A36,Def6; hence psi.y <= q.u by A31,A35,A39,A40,AXIOMS:25; end; Lm2: the RLSStruct of V is RealLinearSpace proof A1: 0.the RLSStruct of V = the Zero of V by RLVECT_1:def 2 .= 0.V by RLVECT_1:def 2; A2: now let v',w' be VECTOR of V, v,w be VECTOR of the RLSStruct of V; assume A3: v = v' & w = w'; hence v + w = (the add of V).[v',w'] by RLVECT_1:def 3 .= v' + w' by RLVECT_1:def 3; let r be Real; thus r *v = (the Mult of V).[r,v'] by A3,RLVECT_1:def 4 .= r* v' by RLVECT_1:def 4; end; the RLSStruct of V is Abelian add-associative right_zeroed right_complementable RealLinearSpace-like proof hereby let v,w be VECTOR of the RLSStruct of V; reconsider v' = v, w' = w as VECTOR of V; thus v + w = w' + v' by A2 .= w + v by A2; end; hereby let u,v,w be VECTOR of the RLSStruct of V; reconsider u' = u, v' = v, w' = w as VECTOR of V; A4: v + w = v' + w' by A2; u + v = u' + v' by A2; hence (u + v) + w = (u' + v') + w' by A2 .= u' + (v' + w') by RLVECT_1:def 6 .= u + (v + w) by A2,A4; end; hereby let v be VECTOR of the RLSStruct of V; reconsider v' = v as VECTOR of V; thus v + 0.the RLSStruct of V = v' + 0.V by A1,A2 .= v by RLVECT_1:10; end; hereby let v be VECTOR of the RLSStruct of V; reconsider v' = v as VECTOR of V; consider w' being VECTOR of V such that A5: v' + w' = 0.V by RLVECT_1:def 8; reconsider w = w' as VECTOR of the RLSStruct of V; take w; thus v + w = 0.the RLSStruct of V by A1,A2,A5; end; hereby let a be Real, v,w be VECTOR of the RLSStruct of V; reconsider v' = v, w' = w as VECTOR of V; A6: a * v = a * v' & a * w = a * w' by A2; v + w = v' + w' by A2; hence a * (v + w) = a * (v' + w') by A2 .= a * v' + a * w' by RLVECT_1:def 9 .= a * v + a * w by A2,A6; end; hereby let a,b be Real, v be VECTOR of the RLSStruct of V; reconsider v' = v as VECTOR of V; A7: a * v = a * v' & b * v = b * v' by A2; thus (a + b) * v = (a + b) * v' by A2 .= a * v' + b * v' by RLVECT_1:def 9 .= a * v + b * v by A2,A7; end; hereby let a,b be Real, v be VECTOR of the RLSStruct of V; reconsider v' = v as VECTOR of V; A8: b * v = b * v' by A2; thus (a * b) * v = (a * b) * v' by A2 .= a * (b * v') by RLVECT_1:def 9 .= a * (b * v) by A2,A8; end; let v be VECTOR of the RLSStruct of V; reconsider v' = v as VECTOR of V; thus 1 * v = 1 * v' by A2 .= v by RLVECT_1:def 9; end; hence thesis; end; Lm3: for V',W' being RealLinearSpace st V' = the RLSStruct of V for W being Subspace of V st W' = the RLSStruct of W holds W' is Subspace of V' proof let V',W' be RealLinearSpace such that A1: V' = the RLSStruct of V; let W be Subspace of V; assume A2: W' = the RLSStruct of W; hence the carrier of W' c= the carrier of V' by A1,RLSUB_1:def 2; thus the Zero of W' = the Zero of V' by A1,A2,RLSUB_1:def 2; thus the add of W' = (the add of V') | [:the carrier of W',the carrier of W':] by A1,A2,RLSUB_1:def 2; thus the Mult of W' = (the Mult of V') | [:REAL, the carrier of W':] by A1,A2,RLSUB_1:def 2; end; Lm4: for V' being RealLinearSpace st V' = the RLSStruct of V for f being linear-Functional of V' holds f is linear-Functional of V proof let V' be RealLinearSpace such that A1: V' = the RLSStruct of V; let f be linear-Functional of V'; reconsider f' = f as Functional of V by A1; A2: f' is additive proof let x,y be VECTOR of V; reconsider x' = x, y' = y as VECTOR of V' by A1; thus f'.(x+y) = f.((the add of V').[x',y']) by A1,RLVECT_1:def 3 .= f.(x'+y') by RLVECT_1:def 3 .= f'.x+f'.y by Def4; end; f' is homogeneous proof let x be VECTOR of V, r be Real; reconsider x' = x as VECTOR of V' by A1; thus f'.(r*x) = f.((the Mult of V').[r,x']) by A1,RLVECT_1:def 4 .= f'.(r*x') by RLVECT_1:def 4 .= r*f'.x by Def5; end; hence f is linear-Functional of V by A2; end; Lm5: for V' being RealLinearSpace st V' = the RLSStruct of V for f being linear-Functional of V holds f is linear-Functional of V' proof let V' be RealLinearSpace such that A1: V' = the RLSStruct of V; let f be linear-Functional of V; reconsider f' = f as Functional of V' by A1; A2: f' is additive proof let x,y be VECTOR of V'; reconsider x' = x, y' = y as VECTOR of V by A1; thus f'.(x+y) = f.((the add of V).[x',y']) by A1,RLVECT_1:def 3 .= f.(x'+y') by RLVECT_1:def 3 .= f'.x+f'.y by Def4; end; f' is homogeneous proof let x be VECTOR of V', r be Real; reconsider x' = x as VECTOR of V by A1; thus f'.(r*x) = f.((the Mult of V).[r,x']) by A1,RLVECT_1:def 4 .= f'.(r*x') by RLVECT_1:def 4 .= r*f'.x by Def5; end; hence f is linear-Functional of V' by A2; end; theorem Th35: for V being RealLinearSpace, X being Subspace of V, q being Banach-Functional of V, fi being linear-Functional of X st for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v ex psi being linear-Functional of V st psi|the carrier of X=fi & for x being VECTOR of V holds psi.x <= q.x proof let V be RealLinearSpace, X be Subspace of V, q be Banach-Functional of V, fi be linear-Functional of X; assume A1: for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v; defpred P[Element of PFuncs(the carrier of V,REAL)] means ex Y being Subspace of V st the carrier of X c= the carrier of Y & $1|the carrier of X = fi & ex f' being linear-Functional of Y st $1 = f' & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v; reconsider A = { f where f is Element of PFuncs(the carrier of V,REAL): P[f]} as Subset of PFuncs(the carrier of V,REAL) from SubsetD; the carrier of X c= the carrier of V by RLSUB_1:def 2; then fi in PFuncs(the carrier of X,REAL) & PFuncs(the carrier of X,REAL) c= PFuncs(the carrier of V,REAL) by PARTFUN1:119,125; then reconsider fi' = fi as Element of PFuncs(the carrier of V,REAL); fi'|the carrier of X = fi by FUNCT_2:40; then fi in A by A1; then reconsider A as non empty Subset of PFuncs(the carrier of V,REAL); defpred P[set, set] means $1 c= $2; A2: for x being Element of A holds P[x,x]; A3: for x,y being Element of A st P[x,y] & P[y,x] holds x = y by XBOOLE_0:def 10; A4: for x,y,z being Element of A st P[x,y] & P[y,z] holds P[x,z] by XBOOLE_1:1 ; A5: now let B be set such that A6: B c= A and A7: for x,y being Element of A st x in B & y in B holds P[x,y] or P[y,x]; per cases; suppose A8: B = {}; consider u being Element of A; take u; let x be Element of A; assume x in B; hence P[x,u] by A8; suppose A9: B <> {}; A10: B is Subset of PFuncs(the carrier of V,REAL) by A6,XBOOLE_1:1; then reconsider E = B as non empty functional set by A9,Th4; B is c=-linear proof let x,y be set; assume x in B & y in B; hence x c= y or y c= x by A6,A7; end; then reconsider u = union B as Element of PFuncs(the carrier of V,REAL) by A10,Th13; consider t being Element of B; A11: now let t be set; assume t in A; then consider f being Element of PFuncs(the carrier of V,REAL) such that A12: t = f and A13: ex Y being Subspace of V st the carrier of X c= the carrier of Y & f|the carrier of X = fi & ex f' being linear-Functional of Y st f = f' & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v; consider Y being Subspace of V such that A14: the carrier of X c= the carrier of Y and A15: f|the carrier of X = fi and A16: ex f' being linear-Functional of Y st f = f' & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A13; consider f' being linear-Functional of Y such that A17: f = f' and A18: for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A16; take Y; reconsider f = f' as linear-Functional of Y; take f; thus t = f by A12,A17; thus the carrier of X c= the carrier of Y by A14; thus f|the carrier of X = fi by A15,A17; thus for y being VECTOR of Y, v being VECTOR of V st y = v holds f.y <= q.v by A18; end; t in A by A6,A9,TARSKI:def 3; then consider Y being Subspace of V, f' being linear-Functional of Y such that A19: t = f' and the carrier of X c= the carrier of Y & f'|the carrier of X = fi & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A11; A20: dom f' = the carrier of Y by FUNCT_2:def 1; t c= u by A9,ZFMISC_1:92; then A21: u <> {} by A19,A20,RELAT_1:60,XBOOLE_1:3; set K = { dom g where g is Element of E: not contradiction }; A22: dom u = union K by Th5; now let x be set; assume x in { dom g where g is Element of E: not contradiction }; then consider g being Element of E such that A23: dom g = x; g in A by A6,TARSKI:def 3; then consider Y being Subspace of V, f' being linear-Functional of Y such that A24: g = f' and the carrier of X c= the carrier of Y & f'|the carrier of X = fi & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A11; dom g = the carrier of Y by A24,FUNCT_2:def 1; hence x c= the carrier of V by A23,RLSUB_1:def 2; end; then reconsider D = dom u as non empty Subset of V by A21,A22,RELAT_1:64,ZFMISC_1:94; D is lineary-closed proof hereby let v,u be Element of V; assume A25: v in D & u in D; then consider D1 being set such that A26: v in D1 & D1 in K by A22,TARSKI:def 4; consider g1 being Element of E such that A27: D1 = dom g1 by A26; consider D2 being set such that A28: u in D2 & D2 in K by A22,A25,TARSKI:def 4; consider g2 being Element of E such that A29: D2 = dom g2 by A28; g1 in A & g2 in A by A6,TARSKI:def 3; then g1 c= g2 or g2 c= g1 by A7; then consider g being Element of E such that A30: g1 c= g & g2 c= g; A31: D1 c= dom g & D2 c= dom g by A27,A29,A30,RELAT_1:25; g in A by A6,TARSKI:def 3; then consider Y being Subspace of V, f' being linear-Functional of Y such that A32: g = f' and the carrier of X c= the carrier of Y & f'|the carrier of X = fi & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A11; A33: dom g = the carrier of Y by A32,FUNCT_2:def 1; then v in Y & u in Y by A26,A28,A31,RLVECT_1:def 1; then v + u in Y by RLSUB_1:28; then v + u in dom g & dom g in K by A33,RLVECT_1:def 1; hence v + u in D by A22,TARSKI:def 4; end; let a be Real, v be Element of V; assume v in D; then consider D1 being set such that A34: v in D1 & D1 in K by A22,TARSKI:def 4; consider g being Element of E such that A35: D1 = dom g by A34; g in A by A6,TARSKI:def 3; then consider Y being Subspace of V, f' being linear-Functional of Y such that A36: g = f' and the carrier of X c= the carrier of Y & f'|the carrier of X = fi & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A11; A37: dom g = the carrier of Y by A36,FUNCT_2:def 1; then v in Y by A34,A35,RLVECT_1:def 1; then a * v in Y by RLSUB_1:29; then a * v in dom g & dom g in K by A37,RLVECT_1:def 1; hence a * v in D by A22,TARSKI:def 4; end; then consider Y being strict Subspace of V such that A38: the carrier of Y = dom u by RLSUB_1:43; consider t being Element of dom u; consider XX being set such that A39: t in XX & XX in K by A22,A38,TARSKI:def 4; consider gg being Element of E such that A40: XX = dom gg by A39; gg in A by A6,TARSKI:def 3; then consider YY being Subspace of V, ff being linear-Functional of YY such that A41: gg = ff and A42: the carrier of X c= the carrier of YY and A43: ff|the carrier of X = fi and for y being VECTOR of YY, v being VECTOR of V st y = v holds ff.y <= q.v by A11; A44: XX = the carrier of YY by A40,A41,FUNCT_2:def 1; XX c= dom u by A22,A39,ZFMISC_1:92; then A45: the carrier of X c= the carrier of Y by A38,A42,A44,XBOOLE_1 :1; A46: the carrier of X c= dom ff by A42,FUNCT_2:def 1; ff c= u by A41,ZFMISC_1:92; then A47: u|the carrier of X = fi by A43,A46,Th2; ex f being Function st u = f & dom f c= the carrier of V & rng f c= REAL by PARTFUN1:def 5; then reconsider f' = u as Function of the carrier of Y, REAL by A38,FUNCT_2:def 1,RELSET_1:11; reconsider f' as Functional of Y; A48: f' is additive proof let x,y be VECTOR of Y; consider XXx being set such that A49: x in XXx & XXx in K by A22,A38,TARSKI:def 4; consider ggx being Element of E such that A50: XXx = dom ggx by A49; A51: ggx in A by A6,TARSKI:def 3; consider XXy being set such that A52: y in XXy & XXy in K by A22,A38,TARSKI:def 4; consider ggy being Element of E such that A53: XXy = dom ggy by A52; ggy in A by A6,TARSKI:def 3; then ggx c= ggy or ggy c= ggx by A7,A51; then consider gg being Element of E such that A54: gg = ggx or gg = ggy and A55: ggx c= gg & ggy c= gg; gg in A by A6,TARSKI:def 3; then consider YY being Subspace of V, ff being linear-Functional of YY such that A56: gg = ff and the carrier of X c= the carrier of YY & ff|the carrier of X = fi & for y being VECTOR of YY, v being VECTOR of V st y = v holds ff.y <= q.v by A11; A57: dom ff = the carrier of YY by FUNCT_2:def 1; dom ggx c= dom gg & dom ggy c= dom gg by A55,RELAT_1:25; then reconsider x' = x, y' = y as VECTOR of YY by A49,A50,A52,A53,A56, FUNCT_2:def 1; A58: ff c= f' by A56,ZFMISC_1:92; the carrier of YY c= the carrier of Y by A22,A38,A49,A50,A52,A53,A54,A56,A57,ZFMISC_1:92; then YY is Subspace of Y by RLSUB_1:36; then x + y = x' + y' by RLSUB_1:21; hence f'.(x+y) = ff.(x'+y') by A57,A58,GRFUNC_1:8 .= ff.x' + ff.y' by Def4 .= f'.x+ff.y' by A57,A58,GRFUNC_1:8 .= f'.x+f'.y by A57,A58,GRFUNC_1:8; end; f' is homogeneous proof let x be VECTOR of Y, r be Real; consider XX being set such that A59: x in XX & XX in K by A22,A38,TARSKI:def 4; consider gg being Element of E such that A60: XX = dom gg by A59; gg in A by A6,TARSKI:def 3; then consider YY being Subspace of V, ff being linear-Functional of YY such that A61: gg = ff and the carrier of X c= the carrier of YY and ff|the carrier of X = fi and for y being VECTOR of YY, v being VECTOR of V st y = v holds ff.y <= q.v by A11; A62: dom ff = the carrier of YY by FUNCT_2:def 1; reconsider x' = x as VECTOR of YY by A59,A60,A61,FUNCT_2:def 1; A63: ff c= f' by A61,ZFMISC_1:92; the carrier of YY c= the carrier of Y by A22,A38,A59,A60,A61,A62,ZFMISC_1:92; then YY is Subspace of Y by RLSUB_1:36; then r*x = r*x' by RLSUB_1:22; hence f'.(r*x) = ff.(r*x') by A62,A63,GRFUNC_1:8 .= r*ff.x' by Def5 .= r*f'.x by A62,A63,GRFUNC_1:8; end; then reconsider f' as linear-Functional of Y by A48; now let y be VECTOR of Y, v be VECTOR of V such that A64: y = v; consider XX being set such that A65: y in XX & XX in K by A22,A38,TARSKI:def 4; consider gg being Element of E such that A66: XX = dom gg by A65; gg in A by A6,TARSKI:def 3; then consider YY being Subspace of V, ff being linear-Functional of YY such that A67: gg = ff and the carrier of X c= the carrier of YY and ff|the carrier of X = fi and A68: for y being VECTOR of YY, v being VECTOR of V st y = v holds ff.y <= q.v by A11; A69: dom ff = the carrier of YY by FUNCT_2:def 1; reconsider y' = y as VECTOR of YY by A65,A66,A67,FUNCT_2:def 1; A70: ff c= f' by A67,ZFMISC_1:92; ff.y' <= q.v by A64,A68; hence f'.y <= q.v by A69,A70,GRFUNC_1:8; end; then u in A by A45,A47; then reconsider u as Element of A; take u; let x be Element of A; assume x in B; hence P[x,u] by ZFMISC_1:92; end; consider psi being Element of A such that A71: for chi being Element of A st psi <> chi holds not P[psi,chi] from Zorn_Max(A2,A3,A4,A5); psi in A; then consider f being Element of PFuncs(the carrier of V,REAL) such that A72: f = psi and A73: ex Y being Subspace of V st the carrier of X c= the carrier of Y & f|the carrier of X = fi & ex f' being linear-Functional of Y st f = f' & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v; consider Y being Subspace of V such that A74: the carrier of X c= the carrier of Y and A75: f|the carrier of X = fi and A76: ex f' being linear-Functional of Y st f = f' & for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A73; consider f' being linear-Functional of Y such that A77: f = f' and A78: for y being VECTOR of Y, v being VECTOR of V st y = v holds f'.y <= q.v by A76; reconsider RLSY = the RLSStruct of Y as RealLinearSpace by Lm2; reconsider ggh = f' as linear-Functional of RLSY by Lm5; A79: f = ggh by A77; reconsider RLS = the RLSStruct of V as RealLinearSpace by Lm2; A80: RLSY is Subspace of RLS by Lm3; A81: now assume the RLSStruct of Y <> the RLSStruct of V; then the carrier of Y <> the carrier of V & the carrier of Y c= the carrier of V by A80,RLSUB_1:40,def 2; then the carrier of Y c< the carrier of V by XBOOLE_0:def 8; then consider v being set such that A82: v in the carrier of V & not v in the carrier of Y by RLSUB_2:77; reconsider v as VECTOR of V by A82; consider phi being linear-Functional of Y + Lin{v} such that A83: phi|the carrier of Y = f' and A84: for x being VECTOR of Y + Lin{v}, v being VECTOR of V st x = v holds phi.x <= q.v by A78,A82,Lm1; A85: dom phi = the carrier of Y + Lin{v} by FUNCT_2:def 1; the carrier of Y c= the carrier of Y + Lin{v} by Th14; then A86: the carrier of X c= the carrier of Y + Lin{v} by A74,XBOOLE_1:1; A87: dom phi c= the carrier of V by A85,RLSUB_1:def 2; rng phi c= REAL by RELSET_1:12; then reconsider phi as Element of PFuncs(the carrier of V,REAL) by A87,PARTFUN1:def 5; (the carrier of Y) /\ the carrier of X = the carrier of X by A74,XBOOLE_1:28; then phi|the carrier of X = fi by A75,A77,A83,RELAT_1:100; then phi in A by A84,A86; then reconsider phi as Element of A; A88: dom f' = the carrier of Y by FUNCT_2:def 1; A89: the carrier of Lin{v} c= the carrier of Lin{v} + Y by Th14; v in Lin{v} by RLVECT_4:12; then v in the carrier of Lin{v} by RLVECT_1:def 1; then v in the carrier of Lin{v} + Y by A89; then A90: phi <> psi by A72,A77,A82,A85,A88,RLSUB_2:9; psi c= phi by A72,A77,A83,RELAT_1:88; hence contradiction by A71,A90; end; then reconsider psi as linear-Functional of V by A72,A79,Lm4; take psi; thus psi|the carrier of X = fi by A72,A75; let x be VECTOR of V; thus psi.x <= q.x by A72,A76,A81; end; theorem Th36: for V being RealNormSpace holds the norm of V is absolutely_homogeneous subadditive Functional of V proof let V be RealNormSpace; reconsider N = the norm of V as Functional of V; A1: N is absolutely_homogeneous proof let x be VECTOR of V, r be Real; thus N.(r*x) = ||.r*x.|| by NORMSP_1:def 1 .= abs(r)*||.x.|| by NORMSP_1:def 2 .= abs(r)*N.x by NORMSP_1:def 1; end; N is subadditive proof let x,y be VECTOR of V; N.(x) = ||.x.|| & N.(y) = ||.y.|| & N.(x+y) = ||.x+y.|| by NORMSP_1:def 1; hence N.(x+y) <= N.x+N.y by NORMSP_1:def 2; end; hence thesis by A1; end; theorem for V being RealNormSpace, X being Subspace of V, fi being linear-Functional of X st for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= ||.v.|| ex psi being linear-Functional of V st psi|the carrier of X=fi & for x being VECTOR of V holds psi.x <= ||.x.|| proof let V be RealNormSpace, X be Subspace of V, fi be linear-Functional of X such that A1: for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= ||.v.|| ; reconsider q = the norm of V as Banach-Functional of V by Th36; now let x be VECTOR of X, v be VECTOR of V such that A2: x=v; q.v = ||.v.|| by NORMSP_1:def 1; hence fi.x <= q.v by A1,A2; end; then consider psi being linear-Functional of V such that A3: psi|the carrier of X=fi and A4: for x being VECTOR of V holds psi.x <= q.x by Th35; take psi; thus psi|the carrier of X=fi by A3; let x be VECTOR of V; q.x = ||.x.|| by NORMSP_1:def 1; hence psi.x <= ||.x.|| by A4; end;