# An introduction to linear transformations in Hilbert space by Francis Joseph Murray By Francis Joseph Murray

The description for this booklet, An creation to Linear ameliorations in Hilbert house. (AM-4), should be forthcoming.

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N projections with ranges "11 , • • • , mn respectively, then E 1+ • • • +Fn_ is a projection, if' and only if' EiE. = o if' 1 j. If' E 1+ ••• +~ is a projection, then bi is orthogonal to m j f'or i j and the range of' E 1 + . • • +Fn_ is 2l( m, u ••• u mn) ( u indicating logical sum). + + Let us suppose that E 1+ ••• + Fn_ is a projection. EiEj o f'or i j. Hence there is an f" such that 9~ Let g = Ef. Then g E mj and Eig 9. Hence + + + lgl 2 ~ ((E 1+ ••• +Fn_)g,g) = I~= 1 This contradiction indicates that EiEj =O.

Fn_ is 2l( m, u ••• u mn) ( u indicating logical sum). + + Let us suppose that E 1+ ••• + Fn_ is a projection. EiEj o f'or i j. Hence there is an f" such that 9~ Let g = Ef. Then g E mj and Eig 9. Hence + + + lgl 2 ~ ((E 1+ ••• +Fn_)g,g) = I~= 1 This contradiction indicates that EiEj =O. ~ the other hand if' EiEj =O f'or i j, then (E 1 + ••• + Fn_) = E 1 + ••• +~. Since E 1+ ••• +~ ls self'-adjoint, it is a projection by Lemma.

LEMMA 1 • to n2 • ·since T is linear, T* is also linear. (Cf. gil with gi--+ g and such that gi = Tfi for an f i with If i I ~ n. , which converges weakly to an f with lfl ~ n (Cf. TheoranIII, §2 above). Let gf = Tfi. x,h) Cl:+CD = (g,h). Since (T*)* = T, (b) of Theorem II of Chapter IV, §2 implies Tf = g. Since lfl ~ n, lgl ~ 1, g is in An• Thus ~ is closed. LEMMA 2. Let ·T be a linear transformation fromm n1 to n2 for which T- 1 exists. Let Fin be as in Lemma. IN T- 1 is bounded. Let R have radius r and center g 1• For g E R C Rn' we have that g = Tf for an f with If' I ~ n.