# Connexions

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# convexity

Module by: Lee Potter. E-mail the author

A signal may be abstractly represented as an element of a Hilbert space with inner product x,y x y and induced norm x=x,x x x x , where xy x y . A sequence x n x n is said to converge strongly to xx if the sequence of real numbers | x n x| x n x converges to zero. A sequence x n x n is said to converge weakly to xx if limit  n x n ,y=x,y n x n y x y for every y y . Weak convergence implies strong convergence in finite dimensional spaces.

A subset 𝒦 𝒦 is said to be convex if xy𝒦 x y 𝒦 implies that 𝒦𝒦 contains λx+1λy λ x 1 λ y for all λ 0 1 λ 0 1 . A set is closed if it contains all its strong limit points.

A mapping ff between normed linear spaces is said to be nonexpansive if |fxfy||xy| f x f y x y for all xx, yy. In Hilbert spaces, ff is firmly nonexpansive if and only if 2fI 2 f I is nonexpansive. A fixed point of ff is any element in the domain of ff satisfying fx=x f x x .

In Hilbert space, nearest-point projection operators onto convex sets are an important subclass of firmly nonexpansive operators.

## property 1

Let 𝒦𝒦 denote any nonempty, closed, convex subset of a Hilbert space . Then there exists a unique y𝒦 y 𝒦 such that inf z𝒦 {|xz|}=|xy| z 𝒦 x z x y

This correspondence is denoted by y= P 𝒦 x y P 𝒦 x , where P 𝒦 :𝒦 : P 𝒦 𝒦 is said to be the nearest-point projection operator, or simply the projection, of onto the closed convex set 𝒦𝒦. The projection is uniquely characterized by the inequality
y,y𝒦:x P 𝒦 x,y P 𝒦 x0 y y 𝒦 x P 𝒦 x y P 𝒦 x 0
(1)
where · · denotes the real part. The operator P 𝒦 P 𝒦 is linear if and only if 𝒦𝒦 is a subspace.

Results are reported here for Hilbert spaces; most can be directly applied in any uniformly convex Banach space. Consideration of closed convex sets is motivated by the existence and uniqueness of the projection. More generally, a subset 𝒮𝒮 of any Banach space is said to be a Chebyshev set if to each point x x there corresponds a unique point in 𝒮𝒮 closest to xx. Every closed convex subset of a uniformly convex Banach space is a Chebyshev set.

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