Compactness and Topological Equivalence - Applied Functional Analysis

We now turn to two fundamental topological properties of Banach spaces that distinguish finite-dimensional spaces from infinite-dimensional ones: compactness of bounded sets and equivalence of norms.

Compactness¶

In finite dimensions, compactness is characterized by the Heine-Borel theorem: a set is compact if and only if it is closed and bounded.

This picture changes dramatically in infinite dimensions.

Example 2 (The Unit Ball in

\ell^2

is Not Compact)

Consider the Hilbert space $\ell^2$ of square-summable sequences with the standard orthonormal basis $\{e_n\}_{n=1}^\infty$ where $e_n = (0, \ldots, 0, 1, 0, \ldots)$ has a 1 in position $n$ and zeros elsewhere.

Each $e_n$ lies in the closed unit ball since $\|e_n\|_2 = 1$ . However, for any $n \neq m$ :

\|e_n - e_m\|_2 = \sqrt{\|e_n\|_2^2 + \|e_m\|_2^2} = \sqrt{2}

(1)

Every pair of distinct elements is at distance $\sqrt{2}$ apart. Therefore no subsequence of $\{e_n\}$ can be Cauchy, and hence no subsequence converges. The closed unit ball in $\ell^2$ is not compact.

Equivalence of Norms¶

Note that the previous definition calls two norms equivalent when they induce the same underlying topology on $X$ . In other words, they generate the same open sets or intuitively induce the same notion of two elements being close.

Geometric intuition. The condition $a\|x\|^1 \leq \|x\|^2 \leq b\|x\|^1$ says that the unit ball of one norm can be scaled to fit inside the unit ball of the other, and vice versa. In $\mathbb{R}^2$ , for example, the $\ell^1$ ball (a diamond), the $\ell^2$ ball (a circle), and the $\ell^\infty$ ball (a square) all have different shapes — but each one can be scaled up or down to contain, or be contained in, any of the others. This mutual containment of balls is exactly what equivalent topologies means: the same sequences converge, the same sets are open, and the same functions are continuous.