About locally convex space

Question

Is a Banach space a locally convex space? Why?

Recall A locally convex space is a linear topological space in which the topology has a base consisting of convex sets.

Answer 1

It is equivalent to define a space $X$ to be locally convex if there is a collection $\{p_{\alpha}\}$ of seminorms on $X$ so that the topology on $X$ coincides with the initial topology of the seminorms; in particular, the sets

$$\bigcap_{i = 1}^n \{p_{\alpha_i}(x) < \epsilon\}$$

form a base for the neighborhoods of zero.

In the case of a normed space, the family can be taken to consist of the norm (which is also a seminorm); the assumption that the space is Banach isn't necessary.

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The equivalence of the two definitions is discussed here.

Answer 2

In a Banach space (or indeed in any normed space), the topology has a base consisting of open balls, i.e., sets of the form $\{x:\Vert x-a\Vert<\epsilon\}$, and these are convex because of the triangle inequality.