Strictly convex space

Strictly convex spaces are considered in the mathematical sub-area of functional analysis. These are normalized spaces whose norm has certain geometric properties that are important for optimization theory.

Definitions

If a real normalized space, then let the unit sphere, that is the set of all elements with , be the dual space , that is the Banach space of continuous linear functionals with the dual space norm . ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle X_{1}}$ ${\displaystyle x\in X}$ ${\displaystyle \|x\|\leq 1}$ ${\displaystyle X\,'}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$ ${\displaystyle \textstyle \|f\|:=\sup _{x\in X_{1}}|f(x)|}$

A real normalized space is called strictly convex if it fulfills one of the following mutually equivalent conditions: ${\displaystyle (X,\|\cdot \|)}$

• Is for , then there is a real number with . ${\displaystyle \|x+y\|=\|x\|+\|y\|}$ ${\displaystyle x,y\in X\setminus \{0\}}$ ${\displaystyle \lambda }$ ${\displaystyle x=\lambda y}$
• Is for two different , then holds for all real numbers . ${\displaystyle \|x\|=\|y\|=1}$ ${\displaystyle x,y\in X}$ ${\displaystyle \|\lambda x+(1-\lambda )y\|<1}$ ${\displaystyle 0<\lambda <1}$
• Is for two different , then applies . ${\displaystyle \|x\|=\|y\|=1}$ ${\displaystyle x,y\in X}$ ${\displaystyle \|{\tfrac {1}{2}}(x+y)\|<1}$
• The function is strictly convex . ${\displaystyle X\rightarrow \mathbb {R} ,\,x\mapsto \|x\|^{2}}$
• Each accepts the supremum in at most one point. ${\displaystyle f\in X\,'\setminus \{0\}}$ ${\displaystyle X_{1}}$

From the second property it follows directly that the set of extreme points of coincides with the edge of the unit sphere . ${\displaystyle X_{1}}$ ${\displaystyle \partial X_{1}=\{x\in X;\ \|x\|=1\}}$

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

Examples

• Uniformly convex spaces are strictly convex, especially pre-Hilbert spaces and the L ^p spaces for . ${\displaystyle 1<p<\infty }$
• ${\displaystyle \ell ^{1}}$is not strictly convex, because is and , so is . ${\displaystyle x=(1,0,0,\ldots )}$ ${\displaystyle y=(0,1,0,\ldots )}$ ${\displaystyle \|x+y\|=2=\|x\|+\|y\|}$
• Every finite dimensional, strictly convex space is uniformly convex. There are strictly convex spaces that are not uniformly convex; these must then be infinitely dimensional. See also renormalization theorem .

Smoothness

The presented property smoothness (ger .: smoothness ) is the dual to the strict convexity property. It is the correspondence to each the set of Functional with associates. One also calls the duality mapping . According to Hahn-Banach's theorem, is for everyone . A normalized space is called smooth if for each one is one-element. The following sentence now applies: ${\displaystyle F:X\rightarrow {\mathcal {P}}(X\,')}$ ${\displaystyle x\in X}$ ${\displaystyle f\in X\,'}$ ${\displaystyle f(x)=\|x\|^{2}=\|f\|^{2}}$ ${\displaystyle F}$ ${\displaystyle F(x)\not =\emptyset }$ ${\displaystyle x\in X}$ ${\displaystyle F(x)}$ ${\displaystyle x\in X}$

• Be a normalized space. ${\displaystyle X}$

If it is strictly convex, it is smooth. ${\displaystyle X\,'}$ ${\displaystyle X}$

If it is smooth, it is strictly convex. ${\displaystyle X\,'}$ ${\displaystyle X}$

For reflexive spaces one then obtains perfect duality:

• Be a reflexive Banach space . ${\displaystyle X}$

${\displaystyle X\,'}$is strictly convex if and only if is smooth. ${\displaystyle X}$

${\displaystyle X\,'}$is smooth if and only if is strictly convex. ${\displaystyle X}$

Since the duality mapping for smooth spaces only has single-element images, it can also be viewed as a function . One can show that this mapping is continuous if one looks at the standard topology and the weak - * - topology . ${\displaystyle F}$ ${\displaystyle F:X\rightarrow X\,'}$ ${\displaystyle X}$ ${\displaystyle X\,'}$

A renormalization set

In many cases, the standard properties presented here can be obtained by switching to an equivalent standard , because the following applies:

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

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

See also

• Convexity condition : for related classes of normalized spaces

Individual evidence

1. ^ V. Barbu, Th. Precupanu: Convexity and Optimization in Banach Spaces , D. Reidel Publishing Company (1986), ISBN 90-277-1761-3 , sentence 2.13
2. ↑ Peter Kosmol: Optimization and Approximation , Walter de Gruyter (2010), ISBN 3-110-21814-3 , conclusion from Theorem 3.17.1
3. ^ NL Carothers: A short course on Banach space theory , Cambridge University Press (2005), ISBN 0521603722 , chapter 11, page 114
4. ^ V. Barbu, Th. Precupanu: Convexity and Optimization in Banach Spaces , D. Reidel Publishing Company (1986), ISBN 90-277-1761-3 , Theorem 2.6
5. ^ NL Carothers: A short course on Banach space theory , Cambridge University Press (2005), ISBN 0521603722 , Theorem 11.4
6. ^ V. Barbu, Th. Precupanu: Convexity and Optimization in Banach Spaces , D. Reidel Publishing Company (1986), ISBN 90-277-1761-3 , Theorem 2.8
7. ↑ Joram Lindenstrauss : Handbook of the geometry of Banach spaces Volume 1, Elsevier (2001), ISBN 0444828427 , page 33