Homework 7, Real Analysis

Due October 5

Problem 1

Let L^1([a,b]) be the space of real valued continuous functions with the d_1 metric.

  1. The polynomials are dense in L^1([a,b]).
  2. Is L^1([a,b]) separable?

Problem 2

Let f:[a,b]\to\R be a continuous function such that

\displaystyle \int_a^b f(x) x^n dx = 0

for all n=0,1,2,\ldots. Then f(x)=0 for all x\in[a,b].

Problem 3

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

d_{X\times Y} \big( (x_1,y_1), (x_2,y_2) \big) = d_X(x_1,x_2) + d_Y(y_1,y_2).

Problem 4

State whether the following are true:

  1. \overline{A\cup B} \subset \overline{A}\cup \overline{B};
  2. \overline{A\cup B} \supset \overline{A}\cup \overline{B};
  3. \overline{A\cap B} \subset \overline{A}\cap \overline{B}; and
  4. \overline{A\cap B} \supset \overline{A}\cap \overline{B}.

 

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