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Homework 5, Real Analysis

Due date: September 8

Problem 1

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

  1. The polynomials are dense in $latex L^1([a,b])$.

  2. Is $latex L^1([a,b])$ separable?

Problem 2

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

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

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

Problem 3

Let $latex \mathbb S^1$ be the circle and $latex \mathscr A\subset C(\mathbb S^1)$ the algebra of trigonometric polynomials. Then $latex \mathscr A$ separates points.

Problem 4

If $latex f,g\in C(X)$, then $latex \max(f,g), \min(f,g)\in C(X)$ .

Problem 5

If $latex X,Y$ are compact metric spaces, then the tensor space

$latex \displaystyle C(X)\otimes C(Y) = \{ (x,y)\mapsto \sum_{k=1}^n f_k(x)g_k(y): f_k\in C(X), g_k\in C(Y), n\ge1\}$

is dense in $latex C(X\times Y)$.

Note: The product space $latex X\times Y$ has the metric

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