Example of a locally-convex topological vector space which is not metrizable

Question

I seek an example of a locally-convex topological vector space which is not a metric space.

From google I found an example LF-Space. Does there exist other examples ?

Answer 1

Plenty of examples ...
If $X$ is an infinite-dimensional Banach space, then the dual space $X^*$ equipped with the weak-* topology is a locally-convex space which is not metrisable.

Cf Regarding metrizability of weak/weak* topology and separability of Banach spaces. in this context.

Answer 2

The most typical example is the weak topology.

The easy example is $\mathbb{R}^2$ with seminorm defined as $\|(x,y)\| = |x|$, which is not a metric space but only a pseudometric space.

Answer 3

According to $\pi$-base, the set of functions from $[0, 1]$ to $[0, 1]$ with the topology of pointwise convergence is not metrizable: https://topology.pi-base.org/spaces/S000103

It is locally convex by definition: take the seminorms $p_x(f) := |f(x)|$ for all $x$.

I found this example among the answers to this question: Non-Metrizable Topological Spaces