Sublinear function

A sublinear function or sub-linear mapping is in linear algebra a real-valued function on a real or complex vector space, which is positively homogeneous and subadditive. Sublinear functions thus represent a certain generalization of linear functions that need to be homogeneous and additive as each stronger requirements. Each sub-linear function is particularly convex; conversely, every positively homogeneous and convex function is sublinear. Sublinear functions play in the functional analysis in the Hahn- Banach a central role.

Definition

A real-valued function on a vector space over the real or complex numbers is called sublinear if the following two conditions are satisfied for all positive real numbers and for all vectors:

  • ( Positive homogeneity )
  • ( Subadditivity )

The case required homogeneity of degree one. Limitation of the positive real numbers in the definition is important because homogeneous subadditive and for all real numbers and functions are already additive thus linear.

Examples

  • Real-valued linear functions are sublinear; the same is true for the amount real - or complex-valued linear functions.
  • Norms and semi-norms are sublinear; as Minkowski functionals on convex and absorbing quantities.
  • For bounded complex-valued consequences of the Limes superior of the real parts of followers a sublinear mapping.

Properties

Zeros

In zero a sublinear function always has the value zero, what about from the positive homogeneity by setting

Followed. Therefore, the requirement of positive homogeneity can also be extended to the non-negative real numbers. However, a sub-linear function may also have other roots; in particular, the zero function is sublinear.

Positivity and negativity

Sublinear functions can in principle take on negative values ​​. But is at a position, it is necessary on the basis of

Apply at the point where that is. So a sublinear function takes at least as many points to positive values ​​, as it assumes negative values.

Convexity

Each sub-linear function is convex, which directly for real from the subadditivity and positive homogeneity over

Followed. Conversely, every convex function is positive homogeneous and subadditive and hence sublinear what means by setting

Can be shown. In the above definition can thus be replaced by the sub-additive convexity.

Application

An important application of sublinear functions can be found in Hahn- Banach. Thereby demonstrating that a linear functional on a subspace of a real vector space, which is limited by a sublinear function, a linear continuation on the total space, which is also limited by these sublinear function. As a consequence, the Hahn- Banach the existence of a sufficient number of steady and linear functionals on a normed space safe and thus provides a crucial basis for the functional analysis.

Convex set Absorbing set Null vector
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