By Ross Honsberger
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Der zweite Band der linearen Algebra führt den mit "Lineare Algebra 1" und der "Einführung in die Algebra" begonnenen Kurs dieses Gegenstandes weiter und schliesst ihn weitgehend ab. Hierzu gehört die Theorie der sesquilinearen und quadratischen Formen sowie der unitären und euklidischen Vektorräume in Kapitel III.
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Extra info for Linear Algebra Problem Book
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.