Metric Space, Normed Space, and Inner Product space hierarcy

Question

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?

Answer 1

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)