Lemma 6

Tag 02X5, permanent. Cite it as [Bourbaki, Tag 02X5].

Théories spectrales, chapter IV, § 1. Opérateurs compacts sur un espace hilbertien

Let $g$ and $h$ be convex functions defined on intervals I and J of $\mathbf{R}$ , respectively. If $g$ is increasing and defined on the image of $h$ , then the function $g\circ h$ is convex on J.

In fact, for $t\in [0,1]$ and $(x, y)\in J\times J$ , we have

g(h(tx+ (1-t)y))\leqslant g(th(x) + (1-t)h(y))

\leqslant tg(h(x)) + (1-t)g(h(y))

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

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Cites

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