Coercive function

In mathematics , a real-valued function is referred to as coercive (or coercive ) if the function values ​​tend towards positive infinity, if the input values ​​tend towards infinity.

definition

Let be a normalized space and a real-valued function . The function is called coercive if the following applies to all sequences with : ${\displaystyle \left(X,\|\cdot \|\right)}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }\subset X}$ ${\displaystyle \lim \limits _{n\rightarrow \infty }\left\|x_{n}\right\|=+\infty }$

${\displaystyle \lim \limits _{n\rightarrow \infty }f(x_{n})=+\infty }$.

motivation

In general, continuous functions on non- compact sets take no minimum or maximum , e.g. B. does not realize the maximum and the minimum. This function is unlimited upwards and downwards and is not coercive. however, is coercive and takes on the minimum ( ). ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} ,\quad x\mapsto x^{3}}$ ${\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} ,\quad x\mapsto x^{2}}$ ${\displaystyle 0=g(0)}$

The following sentence makes it clear under which conditions a coercive function actually assumes its minimum:

Be a reflexive Banach space and meet at least one of the following conditions: ${\displaystyle X}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$

• ${\displaystyle f}$is weakly semi-continuous from below and coercive
• ${\displaystyle f}$is continuous , convex and coercive

Then accept the minimum. ${\displaystyle f}$

Extension to sesquilinear forms

A complex-valued sesquilinear form is called coercive if the function is real-valued and coercive. This property finds z. B. in the Lemma of Lax-Milgram application. ${\displaystyle B\colon X\times X\rightarrow \mathbb {C} }$ ${\displaystyle x\mapsto B(x,x)}$

The term must not be confused with the coercive field strength .

literature

• Dirk Werner : Functional Analysis . Springer Verlag, 2005. ISBN 3-540-43586-7