Coercive function

In mathematics, a real-valued function as coercive (or coercive ) is called if the function values ​​approach infinity when the input values ​​approach infinity.

Definition

Be a normed space and. The function is called coercive if the following holds for all sequences with:

Motivation

Generally assume continuous functions to non- compact sets no minimum or maximum, for example, the maximum and the minimum is not realized. This function is unbounded downwards and upwards and not coercive. other hand, is coercive and takes the minimum ( ) to.

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

Be a reflexive Banach space and satisfies at least one of the following conditions:

• Is weakly semicontinuous from below and coercive
• Is continuous, convex and coercive

Then, to the minimum.

Extension to Sesquilinearformen

A complex-valued sesquilinear is called coercive if the function is real-valued and coercive. This property is for example in Lemma of Lax - Milgram application.

The term should not be confused with the coercivity.