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.


closed as too broad by Lee Mosher, Watson, Behrouz Maleki, R_D, user99914 Aug 23 '16 at 18:41

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    $\begingroup$ What you are asking for is far too broad, encompassing one or more entire courses in vector calculus, topology, functional analysis. I suggest instead writing a sharp, focussed, and particular question, and when you get that answered maybe write another sharp, focussed, and particular question, and so on. $\endgroup$ – Lee Mosher Aug 23 '16 at 15:20
  • $\begingroup$ Metric space and complete space, to begin with, have nothing to do with linear algebra: they're notions of topology. Banach spaces and Hilbert spaces* are two kinds of topological vector spaces. $\endgroup$ – Bernard Aug 23 '16 at 15:34
  • $\begingroup$ @LeeMosher I read the question differently, namely as asking for recalling and contrasting the definitions. That seems feasible to me. $\endgroup$ – quid Aug 23 '16 at 15:37

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!)

  • $\begingroup$ What is the difference between a complete inner product space and a normal inner product space.Another questions is that are all inner product space norm space or are all normed space inner product space? $\endgroup$ – Absaed Aug 23 '16 at 15:55
  • $\begingroup$ @Absaed Not every inner product space need to be complete, that's the difference. All inner product spaces can be given an (induced) norm but the converse is not true. Some normed space cannot be given an inner product that is compatible with the norm. $\endgroup$ – BigbearZzz Aug 23 '16 at 16:34
  • $\begingroup$ I think there are metric spaces that are also vector spaces with continuous addition and scalar multiplication, whose metrics don't derive from norms. There are certainly norms that don't arise from inner products (see 'parallelogram law'). And there are inner product spaces (necessarily infinite-dimensional) that are not complete. Unfortunately, the one example of the latter species I encountered in my reading, I passed over silently, as if the existence of a non-convergent Cauchy sequence were obvious, which it is not. Young, An Introduction to Hilbert Space, Problem 3.2, gives an example. $\endgroup$ – Calum Gilhooley Aug 23 '16 at 16:44

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