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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- Spring '08
- Topology, #, Metric space, normed vector space, Banach space