By Corneliu Constantinescu
V. 1. Banach areas -- v. 2. Banach algebras and compact operators -- v. three. basic idea of C*-algebras -- v. four. Hilbert areas -- v. five. chosen issues
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Extra resources for C*-Algebras Volume 4: Hilbert Spaces
Sirice (dA(~))AEll is an increasing bounded (decreasing) family in R + ,it follows that p(3) is a Cauchy filter. Hence l i r n a ~ ( x )exists and belongs to B . 2. 1 ( 0 ) Let E be a yre-Hilbert space. 2 A, B subsets of E . a) A' *xly} (0) (the orthogonal set of A ) Let E be a pre-Hilbert space. 3, Z E' for every x E A . Since it follows that A' is a closed vector subspace of E . b) and c) follow from the definition. d) By b) and c), 30 5. 3 ( 0 ) (Pythagoras' Theorem) Let E be a pre-Hilbert space and ( x , ) , ~a~finite family of pairwise orthogonal elements of E .
Take x E E . 1 a 3 F ~ X , -AFXEFI b). 2 a)).
By Steps 2: 3, and 4, (slay + 02) = (NY+ PzIx) = ~ ( Y I x+) P(zIx) = 6 (~1s+ ) = p ( 2 1 ~ )= ~ ( X I Y+)~ ( x I Y ) . Rerr~nrk. This theorem makes it possible to define pre-Hilbert (Hilbert) spaces as norrncd (Banach) spaces satisfying the parallelogram law. 5. 7 ( 0 ) Let E be a pre-Hilbert space and E the completion of the associated normed space. There is a unique scalar product o n E generatzng the n o r m of E . E with this scalar product is called the completion of E . The scalar product of E is the restrzction to E of the scalar product of E.