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hw1 - 1 and C[0 1 the space of continuous functions in the...

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ECE 580, Optimization by Vector Space Methods Assignment # 1 Issued: January 16 Due: January 30, 2008 Reading Assignment: Luenberger , Chapters 1 & 2. Reminder: Class is cancelled Jan. 21 & 23. Problems: 1 Prove that the union of any number of open sets is open, and that the intersection of a finite number of open sets is open 2 Let X be a normed vector space, and let X n denote the vector space consisting of n vectors of the form ( x 1 , . . . , x n ) T . For given y X and x X n , prove that there is a vector a * R n that achieves the minimum, min a bardbl y a T x bardbl 3 The normed vector space X is called strictly normed if bardbl x + y bardbl = bardbl x bardbl + bardbl y bardbl implies that y = θ , or x = αy for some scalar α . (i) Show that L p [0 , 1] is strictly normed for 1 < p < . (ii) Show by example that L p [0 , 1] is not strictly normed for p = 1 or p = . (iii) Show that a * in the previous problem is unique when X is strictly normed. 4 Let L 1 [0 , 1] denote the vector space of integrable functions on [0
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Unformatted text preview: , 1], and C [0 , 1] the space of continuous functions in the supremum norm. Consider the sequence of con-tinuous functions deFned by, x n ( t ) = min(1 , max(0 , 1 − 2 n ( t − 1 2 ))) Is this sequence Cauchy in L 1 [0 , 1]? In C [0 , 1]? Is the sequence convergent in one of these normed vector spaces? 5 ±or any real sequence x prove that lim p →∞ b x b p = b x b ∞ . 6 Read about the Contraction Mapping Theorem, and prove the following corollary: Let S be a closed subset of a Banach space. Let T be a mapping from S to X that is expansive : ±or some constant k > 1, b T ( x ) − T ( y ) b ≥ k b x − y b , x,y ∈ S. Then T has a unique Fxed point. Hint : ±irst prove that the mapping T has an inverse if it is expansive....
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