Can we define inner product on every vector space?
I don't know any example of any vector space that do not have any inner product .
Help me
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Sign up to join this communityCan we define inner product on every vector space?
I don't know any example of any vector space that do not have any inner product .
Help me
If you have a finite dimensional vector space just choose a basis and then define an inner product by assuming that basis is orthonormal. Another way to say the same thing: choosing a basis of a finite dimensional vector space over a field $K$ establishes an isomorphism with $K^n$ where there's a natural inner product.
But I think your question misses an important point. We don't find inner products on vector spaces at random. They come to us because they provide useful information that comes essentially from the source of the vector space itself. You've tagged your question "functional analysis". There you regularly encounter inner products that help you do functional analysis - for example, for Fourier analysis.