Understanding Hilbert Space: A Guide to Bilinear and Quadratic Forms

[benorin]

OK, so I've been there before, Hilbert Space that is. You know, infinite dimensional function space. At least I thought I had, that is until I started reading A Hilbert Space Problem Book by Halmos. So operator theory, right.

What's are bilinear, sesquilinear, conjugate linear, ect. - functionals or forms?

What's a quadradic form?

Someone, anyone, please help me: I'm lost in Hilbert Space.

Edit:I understand inner product spaces, are these things definable in such terms?

 

[HallsofIvy]

A quick google got me this (on borgfinder.com)
"In mathematics, a real linear transformation f from a complex vector space V to another is said to
be antilinear (or conjugate-linear or semilinear) if :f(cx+dy)={c*}f(x)+{d*}f(y) for all c, d in C
and all x, y in V. " c* is the complex conjugate of c.
In mathematics, a sesquilinear form on a complex vector space V is a map V ×× V →? C that is
linear in one argument and antilinear in the other. (The name originates from the numerical prefix
sesqui- meaning "one and a half".) Compare with a bilinear form, which is linear in both
arguments. “

 

[homology]

try starting from Functional Analysis by Lax. I'm using it this semester and its pretty good.

 

[dextercioby]

There's a book by Debnath & Mikusinski :"Introduction to Hilbert Spaces with Applications".

Daniel.

 

[masudr]

What's are bilinear, sesquilinear, conjugate linear, ect. - functionals or forms?

Bilinear means linear in both slots, sesquilinear means linear in one slot and not in the other, conjugate linear means linear in the second slot and the first slot requires conjugation.

A functional is some animal that takes functions for arguments.

 

[masudr]

Edit:I understand inner product spaces, are these things definable in such terms?

A vector space with an inner product defined on it (which satisfies the requirements for inner products -- look in Wikipedia for that) gives you an inner product space. A vector space with a distance function defined on it (which satisfies the required properties -- again see Wikipedia) is called a metric space.

A Hilbert space is a particular case of a inner product space (the inner product denoted by $\langle a | b \rangle$) where the norm of a vector is defined by the inner product so that

$$\|a\|^2=\langle a | a \rangle$$

(in fact we always want the positive root of the RHS). Further, the distance function between two points $a, b$ is defined to be $\|b-a\|$ (where we have associated every vector with a point; remember this does not necessarily have anything to do with Euclidean space or $\mathbb{R}^n$ or $\mathbb{C}^n$ or whatever, but clearly resembles some of their features).That's all there is to it.

 

[benorin]

Thank you all for your responses. I have some specific questions now; I will post them after my analysis homework deadline.