Existence of Maximums and Minimums of Functions on Metric Spaces

Subjects: Metric and Normed Spaces
Links: Continuity on Metric Spaces, Compactness in Metric Spaces

Let $f:X\to \bar{\mathbb{R}}$ 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

We say that $f$ reaches its maximum on $X$ if there’s $x_1\in X$ such that

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 $f:K\to \mathbb{R}$ 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 $\mathbb{R}$, 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 $\mathbb{R}$, and $W$ is a normed vector space over $\mathbb{R}$, 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 $g:X\to \bar{\mathbb{R}}$ if there exists a $\delta >0$ 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 $g:X\to \bar{\mathbb{R}}$ if there exists a $\delta >0$ such that $$g(x_0) \ge g(x) \qquad \forall x \in A \cap B_X(x_0, \delta)$$