# Metric Space, Normed Space, and Inner Product space hierarcy

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?

• See this. – David Mitra Dec 24 '13 at 9:27
• 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. – Prahlad Vaidyanathan Dec 24 '13 at 11:32
• @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? – yoshi Dec 24 '13 at 16:17

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).