## Dini’s theorem and related problems

Let be two real numbers and a sequence of continuous functions which converge pointwise to a continuous function .

1. **Dini’s Theorem.** Suppose that the sequence has the property that . Prove that the convergence is uniform.

2. Suppose that every function is increasing. Prove that the convergence is uniform.

3. If every function is convex on , prove that the convergence is uniform on every compact interval in .

## Agreg 2012 Analysis Part 4

**A Fixed Point Theorem**

This parts wishes to extend the next result (which can be used without proof) to infinite dimension.

**Theorem.** (Browuer) Consider a finite dimensional normed vector space. Consider a convex, closed, bounded non-void set. If is a continuous application, then has a fixed point in .

1. In endowed with we consider the following application:

Prove that is continuous with values in the unit sphere of , but does not admit any fixed points.

2. Consider a normed vector space, a closed, bounded non-void subset of and a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)

i) Let . We can cover (which is compact) by a finite number of open balls of radius : with for every . For we define

Prove that is continuous and that there exists such that for we have .

ii) We introduce the application defined by

Prove that for every we have .

## Compact operator maps weakly convergent sequences into strong convergent sequences

Suppose is a compact operator and is a sequence in such that (i.e. converges weakly). Prove that strongly in .

## Fixed point theorem

Suppose is a compact convex set and is a function satisfying . Prove that has a fixed point ( i.e. with ). Read more…