Functional Analysis HW
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چکیده
Proof. I It is straightforward to prove that D is both a normed vector space and an algebra over C. In order to prove that D is a Banach algebra, we check the followings: • The norm ‖ · ‖d onD is complete. Indeed, let { fn }∞ n=1 ⊂ D be a sequence such that ∑ n ‖ fn ‖d < ∞. Then both ∑ n fn and ∑ f ′ n converges uniformly in C([0, 1]) and hence f (x) = ∞ ∑ n=1 fn (x) = ∞ ∑ n=1 ( fn (0) + ∫ x 0 fn (t) dt )
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