Example 3

Tag 07B7, permanent. Cite it as [Bourbaki, Tag 07B7].

Topological Vector Spaces, II, p. 7, § 2. Convex sets

Let $E$ be a vector space and $\|x\|$ a norm on $E$ ; the unit ball $B$ , formed by the points $x$

such that $\|x\| \leq 1$ , is convex since the relations $\|x\| \leq 1, \|y\| \leq 1$ , imply for $0 \leq \lambda \leq 1$ that

\|\lambda x + (1 - \lambda) y\| \leq \lambda \|x\| + (1 - \lambda) \|y\| \leq \lambda + (1 - \lambda) = 1 .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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