Journal of Formalized Mathematics
Volume 14, 2002
University of Bialystok
Copyright (c) 2002 Association of Mizar Users

Bilinear Functionals in Vector Spaces


Jaroslaw Kotowicz
University of Bialystok

Summary.

The main goal of the article is the presentation of the theory of bilinear functionals in vector spaces. It introduces standard operations on bilinear functionals and proves their classical properties. It is shown that quotient functionals are non degenerated on the left and the right. In the case of symmetric and alternating bilinear functionals it is shown that the left and right kernels are equal.

This work has been partially supported by TRIAL-SOLUTION grant IST-2001-35447 and SPUB-M grant of KBN.

MML Identifier: BILINEAR

The terminology and notation used in this paper have been introduced in the following articles [13] [6] [17] [12] [4] [18] [11] [2] [15] [3] [9] [19] [5] [7] [1] [16] [14] [10] [8]

Contents (PDF format)

  1. Two Form on Vector Spaces and Operations on Them
  2. Functional Generated by Two Form when the One of Arguments is Fixed
  3. Two Form Generated by Functionals
  4. Bilinear Forms and their Properties
  5. Left and Right Kernel of Form. Kernel of ``Diagonal''
  6. Bilinear Symmetric and Alternating Forms

Bibliography

[1] Grzegorz Bancerek. Curried and uncurried functions. Journal of Formalized Mathematics, 2, 1990.
[2] Jozef Bialas. Group and field definitions. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[7] Jaroslaw Kotowicz. Monotone real sequences. Subsequences. Journal of Formalized Mathematics, 1, 1989.
[8] Jaroslaw Kotowicz. Quotient vector spaces and functionals. Journal of Formalized Mathematics, 14, 2002.
[9] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[10] Anna Justyna Milewska. The Hahn Banach theorem in the vector space over the field of complex numbers. Journal of Formalized Mathematics, 12, 2000.
[11] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[12] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[13] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[14] Andrzej Trybulec. A Borsuk theorem on homotopy types. Journal of Formalized Mathematics, 3, 1991.
[15] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[16] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Journal of Formalized Mathematics, 2, 1990.
[17] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[18] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[19] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.

Received November 5, 2002

[ Download a postscript version, MML identifier index, Mizar home page]