The First Application: The Averaging Operator -- Compact Groups Acting on Convex Cones -- More Module Actions, Convolutions -- Complexification of Real Representations -- Postscript -- References for this Chapter -- Additional Reading -- ch. 4 Characters -- pt. 1 Characters of Finite Dimensional Representations -- pt. 2 The Structure Theorem of Efin -- Cyclic Modules -- Postscript -- References for this Chapter -- Additional Reading -- ch. 5 Linear Lie Groups -- Preliminaries -- The Exponential Function and the Logarithm -- Differentiating the Exponential Function in a Banach Algebra -- Local Groups for the Campbell -- Hausdorff Multiplication -- Subgroups of A -- 1 and Linear Lie Groups -- Analytic Groups -- The Intrinsic Exponential Function of a Linear Lie Group -- The Adjoint Representation of a Linear Lie Group -- Subalgebras, Ideals, Lie Subgroups, Normal Lie Subgroups -- Normalizers, Centralizers, Centers -- The Commutator Subgroup -- Forced Continuity of Morphisms between Lie Groups

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