SVD: proof of existence

Question

I'm reading "Numerical Linear Algebra" by Lloyd Thefethen. For Singular Value Decomposition proof of existence it starts like this: "Set $\sigma_1=||A||_2$. By a compactness argument, there must be vectors $ v_1 \in C^n$ and $u_1 \in C^m$ with $||v_1||_2=||u_1||_2=1$ and $Av_1=\sigma_1u_1$." What exactly "by a compactness argument" means? I understand it should have something to do with topological compactness, but how?

Answer 1

Let $S = \{x \in \mathbb C^m \mid \|x \|_2 = 1 \}$. Then $\|A\|_2 = \sup_{x \in S} \|Ax\|_2$.

But the function $f:S \to \mathbb R$ defined by $f(x) = \|Ax\|_2$ is continuous, and $S$ is compact, so (by the extreme value theorem) $f$ attains a maximum value at some point $v_1 \in S$.