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