I gave my masters project on "Bilinear forms." It's a transformation from XxX and linear in both coordinates. I wanted to know what problems can be discussed in this?
If you like, you can think about bilinear forms in Banach spaces. Have a look at the paper
María D. Acosta, Julio Becerra-Guerrero, Domingo García and Manuel Maestre The Bishop-Phelps-Bollobás Theorem for bilinear forms Trans. Amer. Math. Soc. 365 (2013), 5911-5932
It is an active area of research, so you will find a number of related publications that will be sufficient for a good Master thesis.
Thank you Vladmir Kadets, But the Journal is not available for me. I also wish to have some work on this.
you can download the paper here:
https://www.dropbox.com/s/v94rzf6p797bjmh/Acosta-2013-BPB-prop.pdf
(the reference will be valid for a couple of days)
Indefinite Metric Spaces. There are a lot of problem related with this direction. For beginning you can try the book of Azizov & Iokhvidov.
The theories of bilinear forms and associated quadratic forms are very important for research efforts in various areas of mathematics, physics etc. One important area is the classification of such bilinear or quadratic forms and this can be done via K-theory and related mathematics.. If V is a finite dimensional vector space over a field F, one can classify bilinear forms B: V x V ->F and quadratic forms q: V --> F and obtain invariants --Grothendieck groups and rings , Witt groups and rings etc. Indeed the studies of quadratic forms in the context of K-theory is also connected with studies of quadrics and homogeneous varieties, chow groups, motivic cohomology, stable homotopy of spheres, Algebraic cobordism theory etc. When F is the field of complex numbers and A is an involutive Banach algebra, (e. g. C*-algebra) one also obtains Witt groups e. t. c . associated with Hermitian forms and these are connected with non-commutative geometry.
So, it depends on your interest and how far you want to go as far as research is concerned in this direction. A basic text is :W. Scharlau: Quadratic and Hermitian Forms. Springer Verlag, Berlin , 1985. Also in the direction of connections of Quadratic forms to K-theory and other theories, there are introductory lecture notes by Alexander Vishik titled: 'Topics in Quadratic forms' as a chapter in the book " Some recent developments in Algebraic K-theory" ICTP lecture Notes series No 23, edited by
E. Friedlander, A. O. Kuku(myself) and C. Pedrini. pages 233 to 274 (2008). The book is the proceedings of an ICTP School in 2007 directed by Friedlander, Pedrini and myself. On connections with non-commutative geometry there is the book " Non commutative geometry by A. Connes 1994.
I feel I should elaborate a bit more on the connection between Bilinear and Quadratic forms for the benefit of the questioner, Prasanth. Assume that the characterisitic of the field F is not equal to 2 (which is the case for the usual fields you may be interested in---real numbers, complex numbers, rational numbers). A quadratic form on a finite dimensional vector space V over F is a map q: V -->F which is the diagonal part of some symmetric bilinear form B_q : V x V --F i.e. q(v) = B_q (v, v). Also, B_q can be constructed from q uniquely. I. e. B_q (u, v) = q(u+v) - q(u) - q(v).