Remark 4

Tag 09IJ, permanent. Cite it as [Bourbaki, Tag 09IJ].

Integration, VIII, p. 9, § 2. Linear representations of groups

When $E$ is a normed space, $U$ is said to be isometric if $\|U(s)\| = 1$ for every $s \in G$ . For this, it suffices that $\|U(s)\| \leq 1$ for all $s \in G$ , because one then has

1 = \|1\| \leq \|U(s)\| \cdot \|U(s^{-1})\|,

whence

\|U(s)\| = \|U(s^{-1})\| = 1

for all

s \in G

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

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