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I. Topological Preliminaries

A topology T in a set X is a class of subsets of X (called open sets) satisfying the following axioms:
the union of any number of open sets is open;
the intersection of any two (or finite number of) open sets is open;
X and the empty set (∅) are open.
K. Chandrasekharan

II. The Haar measure on a locally compact group

We have used the term ‘measure’ for any non-negative, additive, set function which vanishes on the empty set [cf. Course on Integration].
K. Chandrasekharan

III. Hilbert spaces and the spectral theorem

A Banach space over the complex numbers ℂ, or the real numbers ℝ, is a linear space (over ℂ or ℝ), with a norm ‘‖ ‖’ such that the space is complete with respect to the “metric” d(x, y) = ‖xy‖ defined by the norm. [A norm is a function ‘‖ ‖”, which is non-negative, and real-valued, with the properties: (i) ‖ax‖ = |a|·‖x‖, a ∈ ℂ; (ii) ‖x + y‖ ≤ ‖x‖ + ‖y‖; (iii) ‖x‖ = 0 ⇔ x = 0.]
K. Chandrasekharan

IV. Compact groups and their representations

Our aim is to prove the following results on finite-dimensional representations of compact groups. We wish to show (1) that every representation is equivalent to a unitary representation (2) that every representation is completely reducible, (3) the orthogonality relations, and (4) the Peter-Weyl theorem.
K. Chandrasekharan


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