Can Any Inner Product Be Defined in Infinite Dimensional Vector Spaces?

[neelakash]

Hi everyone,
I need a clarification:I read in E. Butkov's book that an inner product may always be imposed on a finite dimensional linear vector space in a variety of ways...Butkov does not explain the point...Can anyone please clarify this?

I wonder what it would be for an infinite dimensional case...As we all know that Hilbert space used in quantum mechanics is an infinite dimensional space. Yet all the books almost inherently define the scalar product in Hilbert space.Is there any hinge in the story?

-Thanks,

Neel

 

[Marcaias]

First of all, a Hilbert space by definition must have an inner product defined on it. (A Hilbert space is a vector space over the real or complex numbers with a complete inner product.)

Secondly, in a finite-dimensional space, if you fix any positive-definite matrix M, then the expression $$\langle x,y \rangle = x^{\dagger}My$$ defines an inner product.

The usual dot product is a special case when M is the identity matrix.

 

[neelakash]

Thank you