Are there any connections between metric space and inner product space

Question

As mentioned in title, are the any connections between inner product space(well, here we talk about only real space) and metric space? I kind of notice that the axioms satisfied by both inner product and metric are almost the same.

Answer 1

Yes, every inner product space is a metric space, with the "Euclidean metric" defined by $$ d(x,y) = \sqrt{\langle x-y, x-y \rangle} $$

Not every metric on a vector space comes from an inner product though (For instance, $l^1$, the space of summable sequences, is one such example)

Answer 2

There is even an intermediate step: in your setting one can prove that

$$\text{inner product}\Rightarrow \text{norm}\Rightarrow \text{metric/distance},$$

where $\Rightarrow$ means "induces". A first, major difference between the above structures is that inner products and norms require a vector space structure, while metrics are defined on sets.