Download PDF by Francis Joseph Murray: An introduction to linear transformations in Hilbert space

By Francis Joseph Murray

The description for this booklet, An creation to Linear variations in Hilbert area. (AM-4), might be forthcoming.

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Extra info for An introduction to linear transformations in Hilbert space

Example text

1. and f = f +f 2 is the resolution of Theorem VI. ex I 2 • The first sentence is the corollary to Theorem X restated. The second sentence is a consequence of Lemma :; and it·s corollary. §6. ORTHONORMAL SEI'S Since St = m... J. also f'ollows f'rom Lemma 3 and its corollary. f' = f' 1 +f' 2 is the resolution of' Theorem VI since that resolution is unique. ti 2 = Icx Ia°' 12 if' and only if' lf'21 2 = 0 or f' = r, € m. If' Theorem XI is applied to Ji as a closed linear manif'old we obtain: THEOREM XII.

PROOF: We note that if' n ~ m and f' E fi then by Lemma 5 2 2 lfl 2 -2 I< rn

6. ORTHONORMAL SEI'S Since St = m... J. also f'ollows f'rom Lemma 3 and its corollary. f' = f' 1 +f' 2 is the resolution of' Theorem VI since that resolution is unique. ti 2 = Icx Ia°' 12 if' and only if' lf'21 2 = 0 or f' = r, € m. If' Theorem XI is applied to Ji as a closed linear manif'old we obtain: THEOREM XII. There exists an orthonormal set S, <1> 1 , <1>2 , • • • such that '1l(S 1 ) = f;. To every f' € f; we can f'ind a sequence a 1 , a 2 , ••• , a°'= (f' •cx) with If' 12 = rcocx=l I acxl 2 and f' = I:~=l acxcx· If' f' "' la 1 , a 2 , • .

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


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