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Compactness

Definition 1: a metric space \(X\) is compact if every sequence in \(X\) has a convergent subsequence. A subset \(M\) of \(X\) is compact if every sequence in \(M\) has a convergent subsequence whose limit is an element of \(M\).

A general property of compact sets is expressed in the following proposition.

Proposition 1: a compact subset \(M\) of a metric space \((X,d)\) is closed and bounded.

Proof:

Will be added later.

The converse of this proposition is generally false.

Proof:

Will be added later.

However, for a finite dimensional normed space we have the following proposition.

Proposition 2: in a finite dimensional normed space \((X, \|\cdot\|)\) a subset \(M \subset X\) is compact if and only if \(M\) is closed and bounded.

Proof:

Will be added later.

A source of interesting results is the following lemma.

Lemma 1: let \(Y\) and \(Z\) be subspaces of a normed space \((X, \|\cdot\|)\), suppose that \(Y\) is closed and that \(Y\) is a strict subset of \(Z\). Then for every \(\alpha \in (0,1)\) there exists a \(z \in Z\), such that

  1. \(\|z\| = 1\),
  2. \(\forall y \in Y: \|z - y\| \geq \alpha\).
Proof:

Will be added later.

Lemma 1 gives the following remarkable proposition.

Proposition 3: if a normed space \((X, \|\cdot\|)\) has the property that the closed unit ball \(M = \{x \in X | \|x\| \leq 1\}\) is compact, then \(X\) is finite dimensional.

Proof:

Will be added later.

Compact sets have several basic properties similar to those of finite sets and not shared by non-compact sets. Such as the following.

Proposition 4: let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces and let \(T: X \to Y\) be a continuous mapping. Let \(M\) be a compact subset of \((X,d_X)\), then \(T(M)\) is a compact subset of \((Y,d_Y)\).

Proof:

Will be added later.

From this proposition we conclude that the following property carries over to metric spaces.

Corollary 1: let \(M \subset X\) be a compact subset of a metric space \((X,d)\) over a field \(F\), a continuous mapping \(T: M \to F\) attains a maximum and minimum value.

Proof:

Will be added later.