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Running head: REAL ANALYSIS
1
Real Analysis
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REAL ANALYSIS
Real Analysis
Problem 1. For i β β, let ππ = (0,β¦, 0,1,0,β¦) be the vector with 1 at the i-th place and zeros
elsewhere. Let (ππ )βπ=1 be a sequence of real numbers and let f : π00 β κ£ be the linear map
uniquely determined by f(ππ ) = ππ for all i β β..
1). Prove that f is continuous with respect to β.ββ (on π00 ) if and only if the sum of ββ
π=1 ππ is
absolutely convergent.
β
First let suppose that f is continuous and let ββ
π...
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