Sublinear function: Difference between revisions

Revision as of 01:56, 13 January 2022

Revision as of 02:03, 13 January 2022

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More generally, if <math>p : X \to \R</math> is a real-valued sublinear function on a (real or complex) vector space <math>X</math> then

<math display=block>q(x) ~:=~ \sup_{|u|=1} p(u x) ~=~ \sup \{p(u x) : u \text{ is a unit scalar }\}</math>

defines a [[seminorm]] on <math>X.</math> if this supremum is always a real number (that is, never equal to <math>\infty</math>).

===Relation to linear functionals===