All Cauchy sequences over $\mathbf {R}$ converge. Does this mean every inner product space over $\mathbf {R}$ is a complete metric space? If not, what is an example a non-Hilbert inner product space?
2026-04-04 09:42:45.1775295765
Are all inner product spaces over the field of the real numbers Hilbert spaces?
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Not every inner product space is a Hilbert space. Every finite-dimensional one is but there are infinite-dimensional ones which aren't complete, say the space of continuous functions on $[0,1]$ with $(f,g)=\int_0^1 f(x)g(x)\,dx$. Some authors call inner-product spaces "pre-Hilbert spaces" for this reason (the completion is a Hilbert space).