# what is compactness argument

I'm working on this paper and I don't know what is meant by compactness argument in the proof of corrollary 4 page 226 which said that:

the function $$\lambda \to \|B-\lambda A\|$$ (where A and B are in L(H) ) is continuous whith $$\lim_{|\lambda|\to\infty}\|B-\lambda A\|=\infty$$ so by compactness argument, there exists $$z_0\in \mathbb{C}$$ s.t $$\|B-z_0 A\|\leq \|(B-z_0 A)+\lambda A\|,\forall \lambda \in \mathbb{C}$$

• In general, "by a compactness argument" means "we can show this with a proof what 'because a particular set is compact' is the core element". May 14 at 7:52

Hint

This follows from the following general fact, that is not difficult and instructive to prove using compactness. A continuous map $$f : E \to \mathbb R$$ where $$E$$ is a finite dimensional normed space such that $$\lim\limits_{\lVert x \rVert \to \infty} \lvert f(x) \rvert =\infty$$ attains its minimum. Remember that a closed bounded subset of a finite dimensional normed space is compact.

Applied to your question, the result implies that $$\lambda \to \lVert B-\lambda A \rVert$$ attains its minimum at a point that we can name $$z_0 \in \mathbb C$$ and therefore

$$\|B-z_0 A\|\leq \|(B-z_0 A)+\lambda A\|,\forall \lambda \in \mathbb{C}.$$

Since $$\lim_{|\lambda|\to\infty}\|B-\lambda A\|=\infty$$, there is some $$M\in(0,\infty)$$ such that$$|\lambda|>M\implies\|B-\lambda A\|\ge\|B\|=\|B-0\times A\|.$$The set $$K=\{\lambda\in\mathbb{C}:|\lambda|\le M\}$$ is compact, and therefore the function $$\begin{array}{ccc}K&\longrightarrow&[0,\infty)\\\lambda&\mapsto&\|B-\lambda A\|\end{array}$$ has a minimum, at some point $$z_0\in K$$. But then, for each $$\lambda\in\Bbb C$$, $$\|B-z_0A\|\le\|B-\lambda A\|$$, since:

• If $$|\lambda|\le M$$, $$\lambda\in K$$, and therefore $$\|B-z_0A\|\le\|B-\lambda A\|$$.
• If $$\lambda>M$$, then $$\|B-\lambda A\|\ge\|B-0\times A\|\ge\|B-z_0A\|$$.
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• why don't we take $K=\{\lambda:|\lambda|\leq M\}$ May 14 at 9:12
• Isn't that what I did? May 14 at 9:21
• no you take z\in C May 14 at 10:57
• I see what what happened. I wrote $\{z\in\Bbb C:|\lambda|\le M\}$. But I meant to write $\{\lambda\in\Bbb C:|\lambda|\le M\}$. I've edited my answer. May 14 at 11:38