Functional Analysis/Preliminaries

This chapter gathers some standard results that will be used in sequel. In particular, we prove the Hahn-Banach theorem, which is really a result in linear algebra. The proofs of these theorems will be found in the Topology and Linear Algebra books.

Set theory
The axiom of choice states that given a collection of sets $$S_i, i \in I$$, there exists a function
 * $$f: I \to \prod_{i \in I} S_i$$.

Exercise. Use the axiom of choice to prove that any surjection is right-invertible.

In this book the axiom of choice is almost always invoked in the form of Zorn's Lemma.

Topology
Exercise. Prove that $$[0, 1] \cap \mathbf{Q}$$ is not compact by exhibiting an open cover that does not admit a finite subcover.

Exercise. ''Let $$X$$ be a compact metric space, and $$f: X \to X$$ be an isometry: i.e., $$d(f(x), f(y)) = d(x, y)$$. Then f is a bijection.''

Exercise. Prove Tychonoff's theorem for finite product without appeal to Axiom of Choice (or any of its equivalences).

By definition, a compact space is Hausdorff.

In particular, a compact metric space is separable.

Exercise. The lower limit topology on the real line is separable but not second-countable.

We remark that the theorem is also true for a locally compact space, though this version will not be needed in the sequel.

Exercise. Use the theorem to prove the set of real numbers is uncountable.

The next exercise gives a typical application of the theorem.

Exercise. ''Prove Peano's existence theorem for ordinal differential equations: Let $$f$$ be a real-valued continuous function on some open subset of $$\mathbf{R}^n$$. Then the initial value problem''
 * $$\dot x = f(x), x(t_0) = x_0$$

has a solution in some open interval containing $$t_0$$. (Hint: Use Euler's method to construct a sequence of approximate solutions. The sequence probably does not converge but it contains a convergent subsequence according to Ascoli's theorem. The limit is then a desired solution.)

Exercise. ''Deduce Picard–Lindelöf theorem from Peano's existence theorem: Let $$f$$ be a real-valued locally Lipschitz function on some open subset of $$\mathbf{R}^n$$. Then the initial value problem''
 * $$\dot x = f(x), x(t_0) = x_0$$

has a "unique" solution in some open interval containing $$t_0$$. (Hint: the existence is clear. For the uniqueness, use Gronwall's inequality.)

Linear algebra
The theorem means in particular that every vector space has a basis. Such a basis is called a Hamel basis to contrast other bases that will be discussed later.

We remark that a different choice of $$c$$ in the proof results in a different extension. Thus, an extension given by the Hahn-Banach theorem in general is not unique.

Exercise ''State the analog of the theorem for complex vector spaces and prove that this version can be reduced to the real version. (Hint: $$Re f (ix) = Im f (x)$$)''

Note the theorem can be formulated in the following equivalent way.

Exercise. Prove Carathéodory's theorem.

(TODO: mention moment problem.)

Exercise. Given an exact sequence
 * $$0 \to V_1 \to V_2 \to ... \to V_n \to 0$$,

we have: $$(-1)^k \operatorname{dim} V_k = 0$$