Why does it mean for Banach space to be a locally convex topological vector spaces

Question

A Banach space is simply a complete normed linear space. According to Wikipedia it is also a locally convex topological vector space.

How does complete + normed + linear space translate into locally convex + topological vector space?

Answer 1

The norm naturally provides a topology. The triangle inequality implies that the unit ball is convex, for if $\|x\|,\|y\| < 1$ then $\|tx + (1-t)y\| \le t\|x\| + (1-t) \|y\| < 1$. Thus a normed linear space is locally convex.

Answer 2

The norm induces a topology on the vector space in which addition and scalar multiplication are continuous operations. This is the jist of a topologica vector space.

A locally convex vector space is one in which there is a basis of neighborhoods which are convex as subsets of the linear space. (They contain the line segment between any two points, where the concept of a line segment comes from the vector space structure.)

Being normed produces this, because of the triangle inequality the balls of radius $\epsilon$ are convex.

Note also that the a topological vector space is homogeneous, meaning that all points look the same (formally, translation provides a homeomorphism between any two points), so local topological properties can be stated in terms of neighborhoods of the origin.