Existence of Maximums and Minimums of Functions on Metric Spaces

Let $f : X \to \overset{―}{R}$ for the rest of this section

Def: We say that $f$ reaches its minimum on $X$ if there’s an $x_{0} \in X$ such that

f (x_{0}) \leq f (x) \forall x \in X

We say that $f$ reaches its maximum on $X$ if there’s $x_{1} \in X$ such that

f (x_{1}) \geq f (x) \forall x \in X

The point $x_{0}$ is called a minimum of $f$ on $X$ and $x_{1}$ is called a maximum of $f$ on $X$ .

Prop: If $K$ is a non empty compact metric space, then every continuous function $f : K \to R$ reaches its maximum and minimum on $K$

Any two norms in a finite dimensional vector space are equivalent.

With this we can get, the most general form of Heine-Borel. Let $V$ be a finite dimensional vector space over $R$ , then $K$ is a compact subset of $V$ iff $K$ is closed and bounded.

We get that If $V$ is a finite dimensional normed vector space over $R$ , and $W$ is a normed vector space over $R$ , then if $L : V \to W$ is a linear map is Lipschitz continuous.

Def: A subset $A$ of subset of a metric space $X$ . We say that $x_{0} \in A$ is a local minimum of $g : X \to \overset{―}{R}$ if there exists a $δ > 0$ such that $$g(x_0) \le g(x) \qquad \forall x \in A \cap B_X(x_0, \delta)$$We say that $x_{0} \in A$ is a local maximum of $g : X \to \overset{―}{R}$ if there exists a $δ > 0$ such that $$g(x_0) \ge g(x) \qquad \forall x \in A \cap B_X(x_0, \delta)$$