MTH405 Assignment 1 - Assingment 1(1 For 1 p < r < show that `p `r Also show that the inclusion is proper(2 Show that for 1 p < c00 `p c0 c ` That is

MTH405 Assignment 1 - Assingment 1(1 For 1 p < r < show...

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Assingment 1(1) For 1p < r <, show thatpr. Also show that the inclusion is proper.(2) Show that for 1p <,c00(p(c0(c(. That is, show the inclusions are proper. Herec00denotes the set of all finitely supported sequences, andcis the set of all sequences convergent to 0.(3) Find examples of subspaces ofand2which are not closed inand2.(4) Suppose||.||is a norm on a vector spaceXandd(x, y) is the metric induced by this norm. Define anew metricd1(x, y) onXasd1(x, y) = 0 forx=yandd1(x, y) = 1 +d(x, y) forx6=y. Does thereexists a norm that induces this new metric? Justify.(5) This example here shows that there exists infinite-dimensional normed linear space and norms onit which are not equivalent.Consider the normed linear space of all polynomials with complexcoefficients defined on the interval [0,1]. Define the following norms on it. For

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