Example 1

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

Topological Vector Spaces, II, p. 54, § 7. Extremal points and extremal generators

In the space $\mathbf{R}^n$ , all the points of the sphere $S_{n-1}$ are extremal points of the closed ball $B_n$ . For, if $\sum_i y_i^2 \leq 1,\ \sum_i z_i^2 \leq 1$ and $0 < \lambda < 1$ , the relation

\lambda^2 \sum_i y_i^2 + (1 - \lambda)^2 \sum_i z_i^2 + 2\lambda(1 - \lambda) \sum_i y_i z_i = 1 = (\lambda + (1 - \lambda))^2

is possible only if

\sum_i y_i^2 = \sum_i z_i^2 = \sum_i y_i z_i = 1.

But this implies

\sum_i (y_i - z_i)^2 = 0

, thus

y_i = z_i

for all

i

, which proves our assertion.

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

Cited by

Nothing.

Cites

Nothing.