What Distinguishes Hilbert Spaces from Euclidean Spaces?

[WWGD]

So we can say that hilbert space is a a inner product vector space which can be of finite or infinite dimension.if it's finite we can call it as eucledian space...?

Another question

Every vector space where inner product is defined is a hilbert space...is it True ?

No; you need to have a specific relationship between the inner-product and the metric:

the inner-product needs to generate the norm ( and so generate the metric which is itself

generated by the norm.). The space has to be complete under this norm, altho a metric space is,

in a sense, as good as a complete metric space, since it has a completion (tho it is a nice exercise to show that the completion preserves the fact that the metric is generated by the inner-product.)

This happens iff the inner-product satisfies the parallelogram law;

out of all $$L^p$$ and $$l^p$$ spaces, only p=2 gives you a Hilbert space.

See, e.g:

http://math.stackexchange.com/questions/294544/parallelogram-law-valid-in-banach-spaces