Download Characterizations of Inner Product Spaces (Operator Theory by Amir PDF

By Amir

Each mathematician operating in Banaeh spaee geometry or Approximation concept is familiar with, from his personal experienee, that the majority "natural" geometrie homes could faH to carry in a generalnormed spaee except the spaee is an internal produet spaee. To reeall the weIl identified definitions, this suggests IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box pleasing: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =

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Extra info for Characterizations of Inner Product Spaces (Operator Theory Advances and Applications)

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Let u s write S a T i f S and T a r e e q u l p o t e n t . C l e a r l y we h a v e t h e following: (0) s a s, (ii) S a T i m p l i e s T = S, and (iii) S a T and T (iv) is an e q u i v a l e n c e r e l a t i o n . (i) a a U implies S a U: i . e . , I t proves t o be convenient t o be a b l e t o p i c k o u t a d i s t i n g u i s h e d c l a s s of s e t s n, c a l l e d c a r d i n a l numbers, such t h a t f o r each s e t S t h e r e e x i s t s a unique n d l such t h a t S and n a r e e q u i p o t e n t .

W to w *: a i . e . , t o wa w i t h t h e o p p o s i t e o r d e r i n g ) . C h o s e a p o i n t xosX,*. PROOF. S i n c e X,* is i n f i n i t e c o f i n a l s u b s e t of X of m i n i m a l c a r d i n a l n u m b e r , t h e r e i s a g r e a t e r e l e m e n t x , E X , * . C o n t i n u e c h o o s i n g g r e a t e r e l e m e n t s of X D * , u s i n g t h e axiom of choice a n d i n d u c t i o n o n a l o w e r - s a t u r a t e d s u b s e t of On. s u b s e t of X o * , t h a t is c o f i n a l i n X,*, T h i s s u b s e t has a i n t h i s way.

By t h e same argument as used t o e s t a b l i s h (01, one can show t h a t E 6 wY, E 6 w l \ ( x ) , E 6 w ~ ( ~ f) o, r a l l XEX. Let v be t h e l e a s t element i n On - E. R. S i n c e ICE, v 2 2. and 32 v is never a l i m i t o r d i n a l . (1) PROOF. 40 Norman L . A l l i n g By assumption E is non-empty. Let 1eE; then 1 Suppose, f o r a m m e n t , t h a t v is a l i m i t o r d i n a l . 0. s u b s e t s of X , f o r which L < t h e power of L and of R is less t h a n w < {XI < R: E ; which i s a b s u r d .

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