An Introduction to Linear Transformations in Hilbert Space. by Francis Joseph Murray

By Francis Joseph Murray

The description for this publication, An advent to Linear modifications in Hilbert house. (AM-4), could be forthcoming.

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Extra resources for An Introduction to Linear Transformations in Hilbert Space. (AM-4) (Annals of Mathematics Studies)

Example text

Let Tn be a sequence of continuous additive functions, whose dornain is the full space n and whose values lie in a linear space. Suppose that for every f in Tif is convergent. Then the bounds of Tn 1 s are bounded. n, PROOF: Let us suppose that the Theorem does not hold fo~ a specific sequence {Tnl· Then Lemma. re unbounded in every sphere. Now suppose that for i = 1, ••• , k we have specified a function Tn , a sphere ~i' with a center f i and radius ri ~d such t~t i f f E ~· l'rnifl ~ i. Suppose also that ri ~ 2 ri-l ~ 1/2 and ~i+l C ~· We know that the Tnf 's are not bounded in the sphere with center fk and radius ~rk.

Since Tnk is continuous, we can find a closed sphere ~k+l with ceni~r fk+l and radius rk+l ~ ~rk ~ 1 /2k+l for which ITilk+/ I ~ k+1 for f E ~k+l. If g is in ~k+l, lfk-gl ~ lfk-fk+1+fk+1-gl ~ lfk-fk+1 l+lfk+1-gl ~ rk or g in ~k. Hence ~k+ 1 C ~k and we see that we may define a sequence of Tni' ~i' fi' ri, which have the properties given in the preceding paragraph for every i. Since each ~i contains all that follow and rn---+ o as n-+ oo, the fn 's form a convergent sequence, whose limit f is in every sphere ~i.

Thus given n, we take the f'irst n of' the ~a 's, and Lemma 2 of' the preceding section tells us that these are linearly independent. a 1,a2 , ••• l ror which there ls an N such that lf' n ~ N, ~ = o. •. l then f'n € U(S 1 ) and ff'-f'nf E. Thus 12 = mes,) and Lemma 1 now implies the separability postulate. To show the completeness, we must consider a sequence of' elements (f'nl such that ff'n-fml - - o. : 1 fbaf 2 < < and ff'n-gf -+ o. Now if' f'n = l~, 1 ,~, 2 , ••• ~,p' 0, ••• j. Then f'or 11 ) o.

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