By Corneliu Constantinescu
V. 1. Banach areas -- v. 2. Banach algebras and compact operators -- v. three. common thought of C*-algebras -- v. four. Hilbert areas -- v. five. chosen themes
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Extra resources for C*-Algebras Volume 4: Hilbert Spaces
Sample text
4, ( x , ) , ~is~summable. Since for every y E, x - xL E I' 2 = -xx. LEI E x , LEI = 2 . 1 ( 0 ) Let E be a pre-Hilbert space, F a complete subspace of E and take (x, y) E E x F . Then the following are equivalent: If these conditions hold, then a 3 b . Take z E F and cw E I K . 2 that 0 I re(x - yly - (y - a z ) ) = re(x - ylcwz) = reti(x - ylz) If we set a := -(x - ~ I z ) then we get successively 0 I -I(x - ylz)I2, ( x - ylz) = 0 , X-YE FI. b + a . Take z ~ F . 2. Now we prove the last assertion.
Take x E E and let A be a nonempty convex set of E . 2 Orthogonal Projections of Hilbert Space A is conlplete with respect to the induced metric, then there is a unique y E A such that y is characterized by the property z EA r e ( x - yly - z ) 1 0 W e define Step 1 Uniqueness Take y, z E A with 112 - yll = llx - zll = d a ( x ) . 1 a ) , Ily - z1I2 < 2112 - yll + 2112 - zll - 4da(z) = 0 Hence Step 2 Existence Let ( x , ) , , ~ b e a sequence in A with lim Ilx, - xi1 = d ~ ( x ) . 1 b ) , ( x , ) , ~is~ a Cauchy sequence in A .
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 . T h e n LEI LEI First consider I = { l , 2 ) . 4, Now let I be arbitrary. We prove the relation by complete induction with respect to Card I . Take X E I and put J := I\(X). 4 ( 0 ) Let E be a Hilbert space and ( x , ) , ~a~family of pairwise orthogonal elements of E . 6. 5 (0) Let E be a Hilberl space and A a subset of E , such that given x, y E A , (xly) = a*,, .