By Francis Joseph Murray

The description for this e-book, An creation to Linear alterations in Hilbert area. (AM-4), could be forthcoming.

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Then since ! 9 1,g 1-g2 1 €I. Our condition implies g 1 = g 2 and thus there is at most one pair If' ,gl €'I with f' in the f'irst place. DEFINITION 4. Let T be a transf'ormation with graph !. If' U(I) is the graph of' a transf'ormation, Ta, this latter transformation is called the additive extension of' T. DEFINITION 5. A transf'ormation T f'rom 1i 1 to fi 2 will be said to be closed, if' its graph is a closed set in Ji1$ 1i 2 • If' [I] is the graph of' a transformation [T], [T ] is called the closure of' T • We note that '1l(I) is the graph of' [Ta] when this latter transf'ormation exiSts.

Oo. This shows that Ef exists f'or every f'. lletric, since for every f' and g of' fi, = I r:=m+1Ecfl2 = (Ef,g) =

This implies that TtT~ ::> (T 2T1 )*. LEMMA 1. d. Let 71* be the set of' f' 's f'or which T*f' = o. Let 7t denote the range of' T. Then 7t • ='1 *. , 7l* is closed. t 10,gl is in 'P if' and only if' g € 7t•. Thus 7t• is the set of' zeros of' T • = -T*. d. and T- 1 and T*-l exist, then (T- 1 )* = T*- 1 • Lemna. 4 of' § 1 and the :;8 IV. ADDTIIVE AND CLOSED TRANSFORMATIONS preceding Lemma. R] and that T- 1 exists if' and only if' [7t*] f:i. = = f:i THH:OREVI VI. R* the range of' T*. Then '1* =7t • , '1 = CR*)•.

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