Due date: September 8 Problem 1 Let L1([a,b]) be the space of real valued continuous functions with the d1 metric. The polynomials are dense in L1([a,b]). Is L1([a,b]) separable? Problem 2 Let f:[a,b]→\R be a continuous function such that ∫baf(x)xndx=0 for all n=0,1,2,…. Then f(x)=0 for all x∈[a,b]. Problem 3 Let S1 be the circle and A⊂C(S1) the algebra of trigonometric polynomials. Then A separates points. Problem 4 If f,g∈C(X), then max . Problem 5 If X,Y are compact metric spaces, then the tensor space \displaystyle C(X)\otimes C(Y) = \{ (x,y)\mapsto \sum_{k=1}^n f_k(x)g_k(y): f_k\in C(X), g_k\in C(Y), n\ge1\} is dense in C(X\times Y). Note: The product space X\times Y has the metric $latex d_{X\times Y} \big( (x_1,y_1), (x_2,y...