17 questions linked to/from Equivalent Definitions of the Operator Norm
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### Operator norm. Alternative definition [duplicate]

Let $T\colon X\to Y$ be a linear operator with norm $$\|T\|=\sup_{\|x\|=1}\|Tx\|.$$ Prove that $$\|T\|=\sup_{\|x\|\leq 1}\|Tx\|.$$
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### Show that two definitions of matrix norm are in fact equivalent $\max\limits_{ x \neq 0} \dfrac{\|Ax\|}{\|x\|} = \max\limits_{\|x\| = 1} \|Ax\|$ [duplicate]

I found a proof in an online course note which purports to show that two definitions of matrix norms are equivalent, however, I have some doubts regarding the proof, I would like a second pair of eyes ...
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### Equivalent definition operator norm [duplicate]

Let $T: X \to Y$ be a bounded linear map between normed spaces. The operator norm is defined by $$\sup_{\|x\| = 1} \|T(x)\|$$ Is this equivalent to $$\sup_{x \in B(0, 1)} \|T(x)\|$$ where $B(0, 1)$...
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I would appreciate it if you could give me a few hints as to how I should go about solving this problem. Suppose $T:\mathfrak{X}\rightarrow \mathfrak{Y}$ is bounded. Show that $\left\|T\right\|=\inf\{... 2answers 5k views ### Proving that the closure of a set contains the$\inf$and$\sup$I came across the following problem about closures: If$A$is a bounded nonempty subset of$\mathbb{R}$, prove that$\sup A \in \overline{A}$and$\inf A \in \overline{A}$. Proof. By hypothesis,$...
In their book on Convex Optimization, Boyd and Vandenberghe state that given a norm, $||\cdot||$, defined on $\mathbb{R}^n$, the dual norm is defined as $$||z||_*= \sup \{ z^Tx : ||x|| \leq 1 \}$$ ...