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I'm having trouble understanding the intuition of the hierarchy of metric space, normed space, and inner product space. What additional structure do I gain at every level? I'm going to list my understanding, I hope others can either fill in more detail or verify that my understanding is correct.

Ok:

  1. A metric space gives me a notion of distance

  2. A normed space introduces a metric to a VS. So it buys me a notion of vector magnitude.

  3. An inner product space enforces a particular norm. This norm, by virtue of being an inner product space, is also a linear functional. I can explicitly leverage my notion of magnitude with my linear functional to build a description of my VS (e.g. compute basis vectors, compute dimension, etc.). I had no way to do this before. Essentially, I have some generic dot product.

is this right?

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  • $\begingroup$ See this. $\endgroup$ – David Mitra Dec 24 '13 at 9:27
  • $\begingroup$ In addition to @DavidMitra's excellent link, note that a metric can be put on any non-empty set, while the others are necessarily vector spaces. $\endgroup$ – Prahlad Vaidyanathan Dec 24 '13 at 11:32
  • $\begingroup$ @DavidMitra good link, thx. I have a question about inner products. So I can define lots of different types of inner products. Like arbitrarily <x,y> = x!*y^2 or something. But the only bilinear forms that satisfy the parallelogram rule form will be a normed space as well? $\endgroup$ – yoshi Dec 24 '13 at 16:17
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You have the right type of ideas, but missing a couple of details that I consider important. Here's how I see it:

To be a metric space, you need no structure on the set, and all you get is a distance with the triangle inequality (in terms of how it has to be constructed) and positive definiteness.

To be either a normed space, or an inner product space, you need a vector space over $\mathbb{R}$ (or $\mathbb{C}$ with some slight adjustments).

A norm gives you a weight which has the triangle inequality in terms of the addition on the space, and respects the multiplication by scalars. This can be used to induced a metric (but not all metrics arise in this way, because there is metrics that don't respect the sums or multiplication).

An inner product gives you the above, except this time the weight is induced by a positive-definite symmetric bilinear form. The fact that it corresponds to this symmetric bilinear form is what gives you the parallelogram identity and other things that exist in inner product spaces but not all normed spaces. (but not all norms arise in this way, because there is norms not induced by a positive-definite bilinear form)

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  • $\begingroup$ could you heuristically explain what you mean by weight? Do you just mean additional stuff I can do at every level? or are you thinking of something else? $\endgroup$ – yoshi Dec 24 '13 at 16:02
  • $\begingroup$ By "weight" I mean a number that measures how far it is from the origin. This is opposed to just knowing how far apart any 2 points are away from each other with no "base point" $\endgroup$ – Matt Rigby Dec 24 '13 at 18:57

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