relation between of uniformly rotund in every direction and uniformly rotund and locally uniformly rotund

Question

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $$\lim_{n→∞}\|x_n−y_n\|=0$$ whenever $$x_n,y_n∈SX$$ are such that $$\lim_{n→∞}\|x_n+y_n\|=2$$ and there is a $z∈X$ and real numbers $λ_n$ such that $$x_n−y_n=λ_nz$$ for all $n∈N$, then we have $$\lim_{n→∞}λ_n=0$$ I want know relation between of uniformly rotund in every direction and uniformly rotund and locally uniformly rotund

Answer 1

First, the fact that UR implies URED is obvious (the definition of UR requires less from the sequences $(x_n)$, $(y_n)$ in order to conclude the convergence $\|x_n-y_n\|\to 0$.)

Second, the notions LUR and URED are not related in general. The answers can be extracted from Chapter II of the monograph by Deville, Godefroy and Zizler Smoothness and renormings of Banach spaces. Let me detail that a little bit.

A norm that is URED does not have to be LUR. An example is $\ell_\infty$ which admits an equivalent URED norm but admits no equivalent LUR norm. Indeed, by Corollary II.6.9 (iii) if there is a countable total set in $X^*$ then $X$ admits an equivalent URED norm. This is the case for $\ell_\infty$ (just take the coordinate functionals). On the other hand, Theorem II.7.10 claims that $\ell_\infty$ does not admit any equivalent norm with Kadec-Klee property (this is a property that is weaker than LUR).

A norm that is LUR does not have to be URED. An example is $c_0(\Gamma)$, where $\Gamma$ is an uncountable set, which admits an equivalent LUR norm (for example the Day norm) but according to Proposition II.7.9 it does not admit any equivalent URED norm.

Answer 2

I will copy here an exercise from Megginson's book An Introduction to Banach Space Theory, since I think it answers at least partially your question.

For definitions on some notions (and also for some other facts related to this notion) you might want to have a look in this book and the references given there.

Exercise 5.32 (A. L. Garkavi, 1962 [84]). A normed space $X$ is uniformly rotund in every direction or uniformly convex in every direction or directionally uniformly rotund if $\delta_X(\epsilon,\to z)$ whenever $0 < \epsilon < 2$ and $z\in S_X$. The abbreviation URED is used for this property.
(a) Prove that a normed space $X$ is (URED) if and only if it has this property: Whenever $(x_n)$ and $(y_n)$ are sequences in $S_X$ such that $\|\frac12(x_n+y_n)\|\to1$ and such that $x_n-t_n\in\langle\{v\}\rangle$ for some $v$ in $X$ and each $n$, it follows that $x_n — y_n \to 0$.
(b) Show that (wUR) $\Rightarrow$ (URED) $\Rightarrow$ (R).
(c) (This uses material from Exercise 5.31). Show that (URWC) $\Rightarrow$ (URED).

Smith gave an example in [219] of a Banach space that is (URED) but not (URWC), and another Banach space that is (R) but not (URED); both examples are formed by equivalently renorming $\ell_2$. It can be shown that a normed space $X$ is (URED) if and only if $\delta_X(\epsilon,\to A) > 0$ whenever $0 < \epsilon < 2$ and $A$ is a nonempty compact subset of $X\setminus\{0\}$; see [218]. Another good source of information on uniform rotundity in every direction is the paper of Day, James, and Swaminathan [57] devoted to the property.

Since uniformly rotund implies weakly uniformly rotund, the part (b) of this exercise answers at least the question about relation of (UR) and (URED). We get that (UR)$\Rightarrow$(URED). Smith's example should be a counterexample for the opposite implication. ((URWC) is defined in the previous exercise, and it fulfills (wUR)$\Rightarrow$(URWC)$\Rightarrow$(R).

References mentioned in this excerpt:

• [57] Mahlon M. Day, Robert C. James, and Srinivasa Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23(1971), 1051-1059, http://dx.doi.org/10.4153/CJM-1971-109-5
• [84] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87-106 (Russian, translated into English as [85]);
• [85] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Amer. Math. Soc. Transl., Ser. 2 39 (1964), 111-132 (English translation of [84]).
• [218] Mark A. Smith, Banach spaces that are uniformly rotund in weakly compact sets of directions, Canad. J. Math. 29 (1977), 963-970; http://dx.doi.org/10.4153/CJM-1977-097-6
• [219] Mark A. Smith, Some examples concerning rotundity in Banach spaces, Mathematische Annalen 1978, Volume 233, Issue 2, pp 155-161, http://dx.doi.org/10.1007/BF01421923