Unitary Representations and Harmonic Analysis: An Introduction

algebra g analytic B₂ C-function called compact group compact set compact supports continuous function converges COROLLARY decomposition defined denoted direct sum dm(x equivalent exp tX finite finite-dimensional formula Fourier series Fourier transform function ƒ function on G ƒ belongs G₁ gl(n H₁ Haar integral Haar measure Hence Hilbert space homomorphism inequality inner product invariant inverse irreducible representations irreducible unitary representation isometry isomorphism L¹(G Lemma Let G Lie algebra Lie group G linear form locally compact Math matrix orthogonal orthonormal basis Parseval equality polynomial positive definite Proof Proposition 3.4 proved q.e.d. Definition q.e.d. Proposition q.e.d. Remark Radon measure real number representation of G satisfies semi-simple seminorms sentation sequence space H strongly continuous subalgebra subset Theorem 1.2 tion topological group trace class transform ƒ uniformly uniquely unitary operator vector space X₁