Is every normed space metrizable?

Question

Is every normed space (infinity dimensional) metrizable ?

Answer 1

Sure, every norm $\lVert\cdot\rVert$ induces a metric via

$$d(x,y)=\lVert x-y\rVert$$

regardless of dimension.

Answer 2

Every norm on a vector space $V$ induces a metric on $V$ by defining $d(x,y) = \|x-y\|$. But of course, not every metric space is of this form; many are not vector spaces and have no norm.

There are also metric vector spaces that are not normable (their topology (though induced by a metric) cannot be induced by a metric from a norm, e.g. $\mathbb{R}^\mathbb{N}$ in the product topology, or the spaces $\ell^p$ for $0 < p <1$.

There also are topological vector spaces, also locally convex ones, that are not metrisable at all (so a fortiori don't have a compatible norm). One example is $\mathbb{R}^{\mathbb{R}}$, the set of all functions from the reals to itself, in the pointwise (aka product) topology.