I do not understand the last line in this picture, why the smallest ball in $H_{2}$ which contains $AS_{1}$ has radius $||A||$, could anyone clarify this for me please?
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1Do you not recognize that this is defining "$||A||$", or do you not recognize why this agrees or disagrees with some other definition of "$||A||$"? – Eric Towers Nov 16 '17 at 03:04
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why this agrees with the definition of the norm given here https://math.stackexchange.com/questions/2521695/the-difference-between-2-definitions-of-the-norm-of-an-operator/2521785#2521785 ?@EricTowers – Nov 16 '17 at 03:08
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Could you prove that $|A| = \inf { c \geq 0 \mid \forall x \in H_1: |Ax| \leq c |x|}$? – Demophilus Nov 16 '17 at 03:18
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@Idonotknow Maybe take a look at this question? – Demophilus Nov 16 '17 at 03:20
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Nitpick: The answer you cite does not contain the norm of an operator (explicitly). (end nitpick) The question it answers does, and both definitions in that question agree with the definition in this question. – Eric Towers Nov 16 '17 at 03:22
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why "both definitions in that question agree with the definition in this question", could you explain please? – Nov 16 '17 at 07:35