What are the difference among Metric space,inner product space,normed space,Euclidean space,Banach space, pre hilbert space, hilbert space and complete space(I only know that they all are belong to vector space)?Please rearrange those terms step by step.I started to learn Linear algebra just a few weeks ago and I have a concept about vector space and inner product space and some others.
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A pre-Hilbert space is the same thing as an inner product space.
Any inner product space is a normed space; and any normed space is a metric space.
A Hilbert space is the same thing as a complete inner product space.
A Banach space is the same thing as a complete normed space.
Therefore, any Hilbert space is a Banach space; and any Banach space is a complete metric space.
All norms on a finite-dimensional (real or complex) vector space are equivalent, so there is essentially only one normed space of each finite dimension; and it is a Hilbert space. (I think this is what the term 'Euclidean space' usually means - but I'm very much open to correction on that point, and indeed on all these points!)