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.) ( αx + βy ) = α ¯ β ( x, y ) ∀ x, y ∈ X and α, β ∈ 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.) αx = α a ∀ x ∈ X and α ∈ 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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## This note was uploaded on 03/14/2011 for the course MATH 676 taught by Professor Staff during the Fall '08 term at Colorado State.

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

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