inverse_lecture7

inverse_lecture7 - 1 Lecture 7 1.1 Definitions Definition 1...

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Unformatted text preview: 1 Lecture 7 1.1 Definitions Definition 1 Let X be a vector space over R (or C ). An inner product is a mapping ( , ) : X X R (or C ) such that (i.) ( x + y, z ) = ( x, z ) + ( y, z ) x, y, z X (ii.) ( x + y ) = ( x, y ) x, y X and , R (or C ) (iii.) ( x, y ) = ( y, x ) x, y X (iv.) ( x, x ) R and ( x, x ) x X (v.) ( x, x ) > if x = 0 Definition 2 Let X be a vector space over R (or C ). A norm on X is a mapping : X R such that (i.) x > x X with x = 0 (ii.) x = a x X and R (or C ) (iii.) x + y x + y x, y X A vector space with norm is called a normed space over R (or C ). Theorem 1 The mapping : X R defined by x = ( x, x ) , x X is a norm. Furthermore, (iv.) | ( x, y ) | x || y x, y X ( Cauchy-Schwartz inequality ) (v.) x y 2 = x 2 + y 2 2 ( x, y ) x, y X (vi.) x + y 2 + x- y 2 = 2 x 2 + 2 y 2 x, y X Definition 3 A sequence...
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inverse_lecture7 - 1 Lecture 7 1.1 Definitions Definition 1...

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