Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I have been studying "Optimization by Vector Space Methods", written by David Luenberger and I am stuck in an obvious point at first glance. My problem is in page 105, where the norm of a linear functional is expressed in alternative ways. The definition for the norm of a linear functional f is (from now on ||y|| denotes the norm of y and |y| the absolute value of y):

||f|| = inf{M: |f(x)| <= M*||x|| for every x in X} where X is the vector space the functional is defined on.

now this definition can be modified and give:

||f|| = inf{M: |f(x/||x||)| <= M for every x in X and x not zero} since f is linear which is equivalent to:

||f|| = sup(|f(x)|/||x||) for x not zero or

||f|| = sup(|f(x)|) for ||x|| = 1

now in the book it states that also:

||f|| = sup(|f(x)|) for ||x|| <= 1

which I can't compehend given the previous definition.

any help is much appreciated,

zok

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# Norm of linear functional

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