# Exercices de mathematiques des oraux de l'ENS. Algebre 1 by Francinou, Gianella, Nicolas. By Francinou, Gianella, Nicolas.

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T(A). Then for each n 0 and g E G, there is some x E A such that y = Then + .. + be one of the seminorms — y = (ir(g'2)x — x)/n. Let 2B/n defining the topology. Then for each n we have where B = sup {flafl a E A). ) Since this is true for all 40 Chapter Two n, — yfl = 0, and since this is true for all seminorms in a sufficient family, 7r(g)y = y for any g E G. 6. Let G be an abelian group acting continuously on a compact metric space X. Then there is a G- invariant probability measure on X. PRooF: M(X) C(X) is compact, convex with the topology.

Let G Iso(E) be a continuous isometric representation of G (where Iso(E) has the strong operator topology). Let A C E be a compact convex G-invariant subset (for the adjoint Chapter Two 42 r of s acting on E) where has the weak-4'-topology. Then there is a C-fixed point in A. Compactness of C will be used only in the following lemma we need for the proof. It says, roughly, that we can take one of the seminorms on E defining the weak-s-topology and make it C-invariant. 4. Fix any x E E. Define hAil0 for A E by hAil0 = Ig C).

REMARK: Although E' is not in general metrizable with the weaktopology even if E is separable, E will be metrizable. 10). Thus, M(X) is actually a compact metrizable space ifX is. 30. Let E be a TVS defined by a sufficient fainis ily of seminorms. Then any element of (E, weakof the form A i-. Afr) for some x E. k is weak-i continuous and linear. E < 1)) is open in E. Hence, there are finitely > 0 such that EEand PRooF: Suppose Then k many elements < 1)) {A E E for all 1 tMzOI < i n}. E with A(xj) = 0 for all i, then = 0 for < 1 for c k.