Sublinear function: Difference between revisions

Revision as of 23:15, 30 January 2023

Revision as of 23:36, 30 January 2023

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{{Math theorem

| name = {{visible anchor|Pryce's sublinearity lemma}}{{sfn|Schechter|1996|pp=313-315}}

| math_statement = Suppose <math>p : X \to \RReals</math> is a sublinear functional on a vector space <math>X</math> and that <math>C \subseteq X</math> is a non-empty convex subset.

If <math>x \in X</math> is a vector and <math>r_0r, RR_0 > 0</math> are positive real numbers such that

<math display=block>p(x) + r_0r RR_0 ~<~ \inf_{c \in C} p\left(x + r_0r c\right)</math>

then for every positive real <math>t > 0</math> there exists some <math>\mathbf{c_0} \in C</math> such that

<math display=block>p\left(x + r_0r \mathbf{c_0}\right) + t RR_0 ~<~ \inf_{c \in C} p\left(x + r_0r \mathbf{c_0} + t c\right).</math>

}}

Adding <math>t RR_0</math> to both sides of the hypothesis <math display=inline>p(x) + r_0r RR_0 \,<\, \inf_{} p\left(x + r_0r C\right)</math> and combining that with the conclusion gives

<math display=block>p(x) + r_0r RR_0 + t RR_0 ~<~ \inf_{} p\left(x + r_0r C\right) + t RR_0 ~\leq~ p\left(x + r_0r \mathbf{c_0}\right) + t RR_0 ~<~ \inf_{} p\left(x + r_0r \mathbf{c_0} + t C\right)</math>

which yields many more inequalities, including, for instance,

<math display=block>p(x) + r_0r RR_0 + t RR_0 ~<~ p\left(x + r_0r \mathbf{c_0}\right) + t RR_0 ~<~ p\left(x + r_0r \mathbf{c_0} + t \mathbf{c_0}\right)</math>

in which an expression on one side of a strict inequality <math>\,<\,</math> can be obtained from the other by replacing the symbol <math>RR_0</math> with <math>\mathbf{c_0}</math> (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).

===Associated seminorm===