Algorithmic Linear Algebra by Herbert Möller

By Herbert Möller

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Let C be a countable dense subset of IK and let 2" be the set of linear combinations of the elements of {(<'lx>yl(x, y) c A x B} with coefficients from C . ~, where 9r denotes the closure of $" in s s F) is dense in s F). 12 d)), it follows t h a t s F) is separable. Now suppose that F is not separable. Since s it follows that s F) is not separable. Hence s F) is isomorphic to F , F) is not separable. Finally, suppose that E is not separable. 2, the map s >s u, > u* is an isometry of real Banach spaces.

18, u E/:I(E) iff u-} E L:2(S), and in this case I1~111 - I1~ -~ I1~. 5 xEA ( 0 ) Let u , v I xEA be positive operators on E such that 6. 4. 6 and in this case ( 0 ) Let u E s u Es IIU* O UII 1 = I iff u*ou E s IIUII 2 . Let A be an orthonormal basis of E . 4. 7 respectively. a) ( 0 ) Let A , B be orthonormal bases of E and F , The map xEA is a scalar product which generates the norm o f / : 2 ( E , F ) . l~>y> = . Ix)yl(x, y) E A x B} is an orthonormal basis o f / : 2 ( E , F ) . d) /f u E / : 2 ( E , F ) , then (x,y)EAxB where the sum is taken in /:2(E, F ) .

If we put (in L2(p| k:=~f(t)y~| then f(~)(-Ix~)y~ (in ~2(L2(u), L2(#))). = ~ tel a & b. 8). Take k E L2(p | u). Then (kxly) = (k(s,t)x(t)du(t))y(s)d#(s) = f ] k(y | x)d(# | u) = (kly | ~) for every (x, y) E 9r" x 6. | by Parseval's equation. 3, kEs IIkll2= Ilkll2,,| Now take u c ~:~(52(u),/2(p)). 2, I(uxly)l ~ < (z,y)~-• ~. 5). Put k :z (x,y)e~'• By the above considerations, (kxl~) = (kly | ~} = (~lv) 52 6. Selected Chapters of C*-Algebras for every (x, y) 9 9v x G, so that N k--u. 12 c)). Then (~ly~> = f(~)~ for all ~, A 9 I.

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