UCED room

Spaces that are uniformly convex in each direction are a class of certain standardized spaces that are examined in the mathematical sub-area of functional analysis. After the English term "uniformly convex in each direction", such spaces are also called UCED spaces or simply UCED.

Definitions

A normed space is known to be uniformly convex if for any two sequences from , and always follows. ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle (x_{n})_{n},(y_{n})_{n}}$ ${\displaystyle \|x_{n}\|\rightarrow 1}$ ${\displaystyle \|y_{n}\|\rightarrow 1}$ ${\displaystyle \|x_{n}+y_{n}\|\rightarrow 2}$ ${\displaystyle \|x_{n}-y_{n}\|\rightarrow 0}$

This property is weakened if convergence is only required if the differences all point in the same direction, more precisely: ${\displaystyle x_{n}-y_{n}}$

A normed space is uniformly convex in the direction if, for any two sequences from , , and always follows. ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle z\in X}$ ${\displaystyle (x_{n})_{n},(y_{n})_{n}}$ ${\displaystyle \|x_{n}\|\rightarrow 1}$ ${\displaystyle \|y_{n}\|\rightarrow 1}$ ${\displaystyle \|x_{n}+y_{n}\|\rightarrow 2}$ ${\displaystyle \{x_{n}-y_{n};\,n\in \mathbb {N} \}\subset \mathbb {R} z}$ ${\displaystyle \|x_{n}-y_{n}\|\rightarrow 0}$

A normalized space is called uniformly convex in every direction or UCED for short if it is uniformly convex in every direction . ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle X}$ ${\displaystyle z\in X}$

Historical remark

The concept of the UCED area was introduced in the investigation of so-called Chebyshev centers. It is the following construction. For a normalized space and two restricted sets , one first defines for ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle H,S\subset X}$ ${\displaystyle s\in S}$

${\displaystyle r_{s}(H):=\sup\{\|s-x\|;\,x\in H\}}$,

this is the maximum distance an element from to . The smallest of these distances is ${\displaystyle H}$ ${\displaystyle s}$

${\displaystyle r(H,S):=\inf\{r_{s}(H);\,s\in S\}}$.

Those for whom this infimum is actually assumed is the so-called Chebyshev Center of in : ${\displaystyle s\in S}$ ${\displaystyle H}$ ${\displaystyle S}$

${\displaystyle {\mathcal {C}}(H,S):=\{s\in S;\,r_{s}(H)=r(H,S)\}}$.

AL Garkavi was interested in standardized spaces in which the Chebyshev center is at most one element of a restricted set in a convex set and thus came to the class of space described here. Indeed, one can show that for every bounded set and every convex set in a UCED space there is at most one element. ${\displaystyle {\mathcal {C}}(H,S)}$ ${\displaystyle H}$ ${\displaystyle S}$

Characterizations

The following statements are equivalent for a normalized space : ${\displaystyle (X,\|\cdot \|)}$

• ${\displaystyle (X,\|\cdot \|)}$ uniformly convex in every direction
• For each and every two sequences it follows from , and always . ${\displaystyle z\in X}$ ${\displaystyle (x_{n})_{n},(y_{n})_{n}}$ ${\displaystyle \|x_{n}\|=\|y_{n}\|=1}$ ${\displaystyle \|x_{n}+y_{n}\|\rightarrow 2}$ ${\displaystyle \{x_{n}-y_{n};\,n\in \mathbb {N} \}\subset \mathbb {R} z}$ ${\displaystyle \|x_{n}-y_{n}\|\rightarrow 0}$
• For each and two episodes with , , and follows . ${\displaystyle z\in X}$ ${\displaystyle (x_{n})_{n},(y_{n})_{n}}$ ${\displaystyle \|x_{n}\|\leq 1}$ ${\displaystyle \|y_{n}\|\leq 1}$ ${\displaystyle \|x_{n}+y_{n}\|\rightarrow 2}$ ${\displaystyle x_{n}-y_{n}\rightarrow z}$ ${\displaystyle z=0}$
• The following applies to all : is and is a consequence in with ${\displaystyle p\in [2,\infty )}$ ${\displaystyle z\in X}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$

${\displaystyle 2^{p-1}(\|x_{n}+z\|^{p}+\|x_{n}\|^{p})-\|2x_{n}+z\|^{p}\rightarrow 0}$,

so follows . ${\displaystyle z=0}$

• There is a such that the following applies: is and is a sequence in with ${\displaystyle p\in [2,\infty )}$ ${\displaystyle z\in X}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$

${\displaystyle 2^{p-1}(\|x_{n}+z\|^{p}+\|x_{n}\|^{p})-\|2x_{n}+z\|^{p}\rightarrow 0}$,

so follows . ${\displaystyle z=0}$

• There is one for all of them , so the following applies: Out , and follows . ${\displaystyle z\in X\setminus \{0\}}$ ${\displaystyle \delta >0}$ ${\displaystyle x\in X}$ ${\displaystyle \|x\|\leq 1}$ ${\displaystyle \|x+z\|\leq 1}$ ${\displaystyle \textstyle \|x+{\frac {1}{2}}z\|<1-\delta }$

Examples

• Uniformly convex spaces are UCED, especially the spaces L ^p ([0,1]) and the sequence spaces for . ${\displaystyle \ell ^{p}}$ ${\displaystyle 1<p<\infty }$
• More generally, even all weakly uniformly convex spaces are UCED.
• The reverse is not true. To do this, define the square-summable sequences on the Hilbert space ${\displaystyle (\ell ^{2},\|\cdot \|_{2})}$

${\displaystyle \|(\xi _{n})_{n}\|_{u}:=\left((|\xi _{1}|+\|(0,\xi _{2},\xi _{3},\ldots )\|_{2})^{2}+\|(a_{2}\xi _{2},a_{3}\xi _{3},\ldots )\|_{2}^{2}\right)^{\frac {1}{2}}}$

where is a zero sequence of positive real numbers. Then UCED is not weakly uniformly convex, not even locally weakly uniformly convex . ${\displaystyle (a_{n})_{n}}$ ${\displaystyle (\ell ^{2},\|\cdot \|_{u})}$

• L ¹ -spaces, L ^∞ -spaces and the function space of the continuous functions on are not UCED. ${\displaystyle C([0,1])}$ ${\displaystyle [0,1]}$

properties

• UCED spaces are strictly convex; the reverse is not true. If you provide something with the norm ${\displaystyle C([0,1])}$

${\displaystyle \|f\|_{s}:=\sup _{t\in [0,1]}|f(t)|+{\sqrt {\int _{0}^{1}|f(t)|^{2}\mathrm {d} t}}}$,

such is a strictly convex Banach space that is not UCED. ${\displaystyle (C([0,1]),\|\cdot \|_{s})}$

• Sub-rooms of UCED rooms are again UCED.

• In UCED spaces, the Chebyshev center is at most one element of a restricted set in a convex set, see above historical remark.

• UCED spaces have normal structure, i.e. every bounded, convex set has normal structure .

Renormalizability

The UCED property can be lost by moving to an equivalent norm . Conversely, therefore, the question arises as to which standardized spaces are isomorphic to a UCED space, i.e. for which standardized spaces there are equivalent norms that make it a UCED space, in short: which spaces are UCED renormalizable.

In this context, the following applies, going back to V. Zizler

• A normalized space is UCED renormalizable if and only if there is an injective, continuous, linear mapping of this space into a UCED space.

This results in the following sentence, which provides examples of UCED renormalizable spaces:

X is isomorphic to a UCED space in the following cases:

• The dual space contains a countable over total quantity, for example if or is a separable space . ${\displaystyle X^{*}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X^{*}}$
• ${\displaystyle X}$is isomorphic to one for any set . ${\displaystyle \ell _{1}(\Gamma )}$ ${\displaystyle \Gamma }$
• ${\displaystyle X}$is isomorphic to a -space for an -finite measure . ${\displaystyle L^{\infty }(\Omega ,\Sigma ,\mu )}$ ${\displaystyle \sigma }$ ${\displaystyle \mu }$

Not all normalized spaces can be renormalized to UCED: they can not be renormalized to UCED for uncountable spaces with discrete topology. ${\displaystyle C_{0}(\Gamma )}$ ${\displaystyle \Gamma }$

See also

• Convexity condition

Individual evidence

1. ↑ AL Garkavi: On the Chebyshev center of a set in a normed space , Investigations of Contemporary Problems in the Constructive Theory of Functions, Moscow (1961), pages 328-331
2. ^ Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity, Theory and Applications , Taylor & Francis Inc. (1983), ISBN 0-8247-1796-1 , sentence 2.6.33. (The formula for (4) or (5) there is incorrect, but the correct formula is proven. The restriction there to Banach spaces is unnecessary.)
3. ^ Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity, Theory and Applications , Taylor & Francis Inc. (1983), ISBN 0-8247-1796-1 , example 2.6.43.
4. ↑ AL Garkavi: The best possible net and the best possible cross-section of a set in a normed space , Izvestia Adad. Naku SSSR (1962), vol. 26, pages 87-106
5. ^ V. Zizler: On some rotundity and smoothness properties of Banach spaces , dissert. Math. Roszprawy (1971), Volume 87, Pages 5-33
6. ↑ MM Day, RC James, S. Swaminathan: Normed Linear Spaces that are Uniformly Convex in Every Direction , Canadian J. Math, (1971), Vol. 23, No. 6, pp. 1051-1059, Theorem 3
7. ↑ MM Day, RC James, S. Swaminathan: Normed Linear Spaces that are Uniformly Convex in Every Direction , Canadian J. Math, (1971), Volume 23, No. 6, pp. 1051-1059, Theorem 2