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The following problem is taken from 'Real Mathematical Analysis' by Pugh, $2$nd edition, page $366,$ exercise $4(a).$

Question: The conorm of a linear transformation $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is $$m(T) = \inf \{ \frac{|Tv|}{|v|}: v \neq 0 \}.$$ It is the minimum stretch that $T$ imparts to vectors in $\mathbb{R}^n.$ Let $U$ be the unit ball in $\mathbb{R}^n.$ (a) Show that the norm and conorm of $T$ are the radii of the smallest ball that contains $TU$ and the largest ball contained in $TU.$

I can use any equivalent definition of $\| T \|.$

What am I supposed to do here? I just need hint on the first part.

Idonknow
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1 Answers1

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Suppose that $A$ is some ball of radius $r$ containing $TU$. Note that $\lVert Tx \rVert \leq r$ for all $x$ with $\lVert x \rVert \leq 1$. Taking the supremum over $x \in U$, will yield the inequality $\lVert T \rVert \leq r$. If $A$ is the smallest such ball, you could argue via contradiction that $r = \lVert T \rVert$. So assume $\lVert T \rVert < r$, but then prove that there's a smaller ball than $A$ that contains $TU$.

A very similar argument can be made for the conorm.

Demophilus
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