Let $T\colon X\to Y$ be a linear operator with norm $$\|T\|=\sup_{\|x\|=1}\|Tx\|.$$ Prove that $$\|T\|=\sup_{\|x\|\leq 1}\|Tx\|.$$
Operator norm. Alternative definition
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functional-analysis
banach-spaces
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3This should be easy. $\{x\in X; \|x\|=1\}\subseteq \{x\in X; \|x\|\le 1\}$ should help you establish one inequality. To get the other one, try to use $\|cx\|=|c|\|x\|$ . – 2011-12-18
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2Also you need to assume that $X\neq\{0\}$ – 2011-12-18