# An introduction to the theory of linear spaces by Shilov G.

By Shilov G.

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The continuity of each π(g) implies π(g)un −→ π(g)u and hence we have π(g)u ∈ Hv1 and Hv1 is invariant. Thus the subrepresentation π1 = π |Hv1 is cyclic with cyclic vector v1 . If Hv1 = H, choose 0 = v2 ∈ Hv⊥1 = H \ Hv1 , consider the closed linear span Hv2 which is π-invariant and orthogonal to Hv1 , and continue like this if H = Hv1 ⊕ Hv2 . Let ξ denote the family of all collections {Hvi }, each composed of a sequence of mutually orthogonal, invariant and cyclic subspaces. We order the family by means of the 38 3.

This Corollary will help to prove the irreducibility in many cases. Here we show the usefulness of Schur’s Lemma by proving another fundamental fact (later we shall generalize this to statements valid also in the inﬁnite-dimensional cases): 20 1. 4: Any ﬁnite-dimensional irreducible representation of an abelian group is one-dimensional. Proof: Let (π, V ) be a representation of G. Then F : V −→ V, v −→ π(g0 )v is an intertwining operator for each g0 ∈ G, as we have π(g)F (v) = π(g)π(g0 )v = π(gg0 )v, as π is a representation, = π(g0 g)v, as G is abelian, = π(g0 )π(g)v, = F (π(g)v).

Representations of Compact Groups The last inequality is a special case of Parseval’s inequality saying that for a unitary matrix representation g −→ A(g) one has k | Ak1 (g) |2 ≤ 1. e. the dimension of H must stay ﬁnite. We already proved that a ﬁnite-dimensional unitary representation is completely reducible. 2: Every unitary representation π of G is a direct sum of irreducible ﬁnitedimensional unitary subrepresentations. 169/170). t. for an arbitrary basis {ei }i∈I of H we have Aei 2 < ∞. i∈I Again, as in the theory of ﬁnite groups, we have ON-relations for the matrix elements of an irreducible representation.