Homework 6, Real Analysis

Due September 28

Problem 1

Let f_n:[a,b]\to\R a monotone sequence of continuous functions which converges pointwise to the continuous function f:[a,b]\to\R. Then f_n\rightrightarrows f on [a,b].

Problem 2

  1. Let K:[0,1]\times[0,1]\to[0,1] be a continuous function and define the operator \mathscr L:C([0,1])\to C([0,1]) by \displaystyle \mathscr Lf(x) = \int_0^1 K(x,y) f(y) dy. Then, the image of the closed ball \bar B_1(0) in C([0,1]) under \mathscr L is compact.
    Such operator is called a compact operator.
  2. Let w:[0,1]\to\R be continuous. Then the operator \displaystyle \mathscr Lf(x) = \int_0^x f(t) w(t)dt is compact.

Problem 3

Let F_1\subset F_2\subset \ldots an increasing sequence of compact subsets of a metric space, and let K \subset\subset \bigcup F_n. Then there exists p such that K\subset F_p.

Problem 4

Let \Omega\subset\R^m be open and f_n:\Omega\to\R an equicontinuous sequence of functions that converges pointwise. Then f_n converges uniformly on each compact subset of \Omega.

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