Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Hahn-Banach Theorem

**by****Bogdan Nowak, and****Andrzej Trybulec**- Received July 8, 1993
- MML identifier: HAHNBAN

- [ Mizar article, MML identifier index ]

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; begin canceled; theorem :: HAHNBAN:2 for X being set, f,g being Function st X c= dom f & f c= g holds f|X = g|X; theorem :: HAHNBAN:3 for A being non empty set, b being set st A <> {b} ex a being Element of A st a <> b; theorem :: HAHNBAN:4 for X,Y being set, B being non empty Subset of PFuncs(X,Y) holds B is non empty functional set; theorem :: HAHNBAN:5 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 }; theorem :: HAHNBAN:6 for A being non empty Subset of ExtREAL st for r being R_eal st r in A holds r <=' -infty holds A = {-infty}; theorem :: HAHNBAN:7 for A being non empty Subset of ExtREAL st for r being R_eal st r in A holds +infty <=' r holds A = {+infty}; theorem :: HAHNBAN:8 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; theorem :: HAHNBAN:9 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; theorem :: HAHNBAN:10 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; canceled; theorem :: HAHNBAN:12 for x,y being R_eal, p,q being real number st x = p & y = q holds (p <= q iff x <=' y); begin :: Chains of Functions definition let A be non empty set; cluster c=-linear non empty Subset of A; end; theorem :: HAHNBAN:13 for X,Y being set for B being c=-linear Subset of PFuncs(X,Y) holds union B in PFuncs(X,Y); begin :: Some Facts on Real Vector Spaces reserve V for RealLinearSpace; theorem :: HAHNBAN:14 for W1,W2 being Subspace of V holds the carrier of W1 c= the carrier of W1 + W2; theorem :: HAHNBAN:15 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]; theorem :: HAHNBAN:16 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; theorem :: HAHNBAN:17 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; theorem :: HAHNBAN:18 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]; theorem :: HAHNBAN:19 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]; theorem :: HAHNBAN:20 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]; theorem :: HAHNBAN:21 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; theorem :: HAHNBAN:22 for V1,V2,W being Subspace of V, W1,W2 being Subspace of W st W1 = V1 & W2 = V2 holds W1 + W2 = V1 + V2; theorem :: HAHNBAN:23 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}; theorem :: HAHNBAN:24 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}; theorem :: HAHNBAN:25 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]; theorem :: HAHNBAN:26 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]; theorem :: HAHNBAN:27 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]; theorem :: HAHNBAN:28 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]; theorem :: HAHNBAN:29 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]; theorem :: HAHNBAN:30 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]; 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 :: HAHNBAN:def 3 for x,y being VECTOR of V holds IT.(x+y) <= IT.x+IT.y; attr IT is additive means :: HAHNBAN:def 4 for x,y being VECTOR of V holds IT.(x+y) = IT.x+IT.y; attr IT is homogeneous means :: HAHNBAN:def 5 for x being VECTOR of V, r being Real holds IT.(r*x) = r*IT.x; attr IT is positively_homogeneous means :: HAHNBAN:def 6 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 :: HAHNBAN:def 7 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 :: HAHNBAN:def 8 for x being VECTOR of V, r being Real holds IT.(r*x) = abs(r)*IT.x; attr IT is 0-preserving means :: HAHNBAN:def 9 IT.(0.V) = 0; end; definition let V; cluster additive -> subadditive Functional of V; cluster homogeneous -> positively_homogeneous Functional of V; cluster semi-homogeneous -> positively_homogeneous Functional of V; cluster semi-homogeneous -> 0-preserving Functional of V; cluster absolutely_homogeneous -> semi-homogeneous Functional of V; cluster 0-preserving positively_homogeneous -> semi-homogeneous Functional of V; end; definition let V; cluster additive absolutely_homogeneous homogeneous Functional of V; 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 :: HAHNBAN:31 for L being homogeneous Functional of V, v being VECTOR of V holds L.(-v) = - L.v; theorem :: HAHNBAN:32 for L being linear-Functional of V, v1,v2 being VECTOR of V holds L.(v1 - v2) = L.v1 - L.v2; theorem :: HAHNBAN:33 for L being additive Functional of V holds L.(0.V) = 0; theorem :: HAHNBAN:34 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; begin theorem :: HAHNBAN:35 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; theorem :: HAHNBAN:36 for V being RealNormSpace holds the norm of V is absolutely_homogeneous subadditive Functional of V; theorem :: HAHNBAN:37 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.||;

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