Strictly convex space

Strictly convex spaces are considered in the mathematical branch of functional analysis. Is standardized spaces, the standard has defined geometric characteristics which are important for the optimization theory.

Definitions

Is a real normed space, so is the unit sphere, ie the set of all elements, is the dual space, ie the Banach space of continuous linear functionals with the dual space norm.

A real normed space is called strictly convex if it satisfies one of the following equivalent conditions with each other:

  • If for, then there's a real number.
  • If for two different, then for all real numbers.
  • If for two different, so true.
  • The function is strictly convex.
  • Each takes the supremum on in at most one point.

From the second property arises directly that the set of extreme points of which coincides with the boundary of the unit sphere.

From the fourth property follows which is important for the optimization theory, stating that a convex set in a strictly convex space at most a minimal vector length.

Examples

  • Uniformly convex spaces are strictly convex, ie in particular pre- Hilbert spaces and Lp- spaces for.
  • Is not strictly convex, as is and so is.
  • Every finite strictly convex space is uniformly convex. There are strictly convex spaces that are not evenly convex; these must then be infinite dimensional. See also Renormierungssatz.

Smoothness

The presented property smoothness (English: smoothness ) is the dual to the strict convexity property. It is the correspondence which assigns to each the set of functionals with. It is also called the duality mapping. By the theorem of Hahn- Banach for all. This is called a smooth normed space if a singleton for each. It is now considered the following sentence:

  • Be a normed space.

For reflexive spaces, we obtain perfect duality:

  • Be a reflexive Banach space.

Since the duality mapping for smooth spaces has only one element pictures, they may be considered as a function. One can show that this map is continuous if we consider the norm topology and the weak -* topology.

A Renormierungssatz

In many cases, one can get by passing to an equivalent norm, the standard properties presented here because it is:

  • Every separable Banach space has an equivalent norm which is both strictly convex and smooth.

In particular, one can construct in this way non- reflexive, strictly convex Banach spaces. This gives you examples of strictly convex but not uniformly convex Banach spaces, because the latter are for a set of Milman always reflexive.

Joram Lindenstrauss
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