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We represent its members by {N a , Ga } where ex runs over a set of indices. Let N be the union of the linear manifolds N a and let G be the bounded linear functional defined over N which is the "union" of the Ga BANACH SPACES 21 in the following sense: If fEN, then f E N~ for some {3 and Gf = G~f. Then for any a we have {N a , Ga } < < {N, G}. This means that there ex~ts in T a pair majorizing every pair of the chain C. Applying Zorn's lemma in one of its favorite forms, we conclude that T contains a maximal element.

Thus p2 = P. Transformations P such that p2 = P will be called idempotent transformations. Note also that M is precisely 4 The pair {M, N} is obviously an ordered pair. In order to lighten the language, we shall omit the participle. 48 SPECTRAL THEORY the set of vectors f such that Pf = f; that N is precisely the set of vectors f such that Pf = o. We recapitulate our findings in THEOREM 6-1: Let {M, N} be a complementary pair of closed linear manifolds in B. Thus, B is the direct sum of M and N, B = M EB N.

It is clear that if T is linear, then the graph of T is a linear manifold of B EB C. " 44 SPECTRAL THEORY THEOREM 4-1: The graph of a linear transformation T is a closed linear manifold if and only if the transformation is closed. Suppose T is a closed transformation. Let {fn, Tfn} , n = 1, 2, . · , be the nth element of a Cauchy sequence in B EB C. 2 Since B EB C is complete, the sequence converges to an element {f, g}. Thus I {f - fn, g - Tfn} I = IIf - fnll + IIg - Tfnll ~ o. Thus fn ~ f and Tfn ~ g.

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Spectral Theory by Edgar Lorch


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