每題20分

	Let T be a linear mapping from the Banach space to the Banach space . Prove that T is bounded iff T is continuous at a point .
	Let ,and ,For each ,the norm of x is defined as . Prove that the sequence converges to 0 weakly and is divergent in norm.
	Let Y be the Banach space which consists of all continuous function defined on the interval [0,1]. For each , defined the norm . Let be a functional such that i) Prove that is continuous. ii) Find a function g which has bounded variation such that .
	Let the Banach space , and k a continuous function defined on . Define the linear operator A on X by . i) Calculate the induced norm . ii) Prove that A is a compact linear operator.
	State　i) open mapping theorem,　　ii) uniformly bounded principle,　iii) Hahn Banach extension theorem, iv) Riesz representation theorem, and　v) Banach contraction mapping principle.