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In an optimization book I am following, the induced norm is defined as the maximum of the norms of the vectors $Ax$ where the vector $x$ runs over the set of all vectors with unit norm.

Now, it says that because of the continuity of a vector norm for each matrix $A$ the maximum $\max_{\left|\left|x\right|\right|=1}\left|\left|Ax\right|\right|$ is attainable, that is, a vector $x_0$ exists such that $\left|\left|x_0\right|\right|=1$ and $\left|\left|Ax_0\right|\right|=\left|\left|A\right|\right|$. The book also says that this follows from the theorem of Weierstrass. I have not understood this part.

Also, it is not clear to me why for any $m \times n$ matrix $A$ and any $x \in \Bbb R^n$ the following inequality gets satisfied :

$$\left|\left|Ax\right|\right|_{(m)} \leq \left|\left|A\right|\right|\,\left|\left|x\right|\right|_{(n)}$$ assuming a above definition of norm.

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The maximum is attained by compactness of the set ${x\in V| \,||x||=1}$ and continuity of the norm and linear maps in finite dimension. (See the extreme value theorem on wikipedia) This is why you can replace $\sup$ (found in most textbooks) by $\max$.

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