# reflexive banach space

I want to ask this non-expert question:

What does it mean geometrically for a Banach space to be reflexive?

Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some other theorems may be. But this doesn't give me a geometric intuition so far.

• I think the question is too broad in its current form but it is certainly closely related to things which are research level mathematics (e.g. uniform convexity, super-reflexivity and the super versions of Radon-Nikodym property and Krein-Milman) Commented Mar 24, 2015 at 14:22
• There is a beautiful result of Odell and Schlumprecht that gives an answer to this question for separable Banach spaces.Odell, E.(1-TX); Schlumprecht, Th.(1-TXAM) Asymptotic properties of Banach spaces under renormings. (English summary) J. Amer. Math. Soc. 11 (1998), no. 1, 175–188. Commented Mar 24, 2015 at 16:12
• A separable Banach space is reflexive iff there is an equivalent norm on the space s.t. whenever $(x_n)$ is a bounded sequence for which $\lim_n \lim_m \|x_n + x_m\| = 2 \lim_n \|x_n\|$, the sequence $(x_n)$ converges. Commented Mar 24, 2015 at 16:15
• Alpx, are you calling me "this non-expert". OK, it's true. The question is interesting but too vague. @BillJohnson 's answer is great. I'd like to see it as THE ANSWER (are there some other "THE ANSERWs"? Then let them see them too). Commented Mar 24, 2015 at 18:31
• OK, I turned it into an answer now that the question has been reopened. Commented Mar 24, 2015 at 19:50

There is a beautiful result of Odell and Schlumprecht that gives an answer to this question for separable Banach spaces.Odell, E.(1-TX); Schlumprecht, Th.(1-TXAM) Asymptotic properties of Banach spaces under renormings. (English summary) J. Amer. Math. Soc. 11 (1998), no. 1, 175–188.

A separable Banach space is reflexive iff there is an equivalent norm on the space s.t. whenever $(x_n)$ is a bounded sequence for which $\lim_n \lim_m \| x_n+x_m \|= 2\lim_n \|x_n\|$, the sequence $(x_n)$ converges.

• Thank you. Now, it's easier to find it and possibly make a reference to it within MO. Commented Mar 24, 2015 at 21:05
• @billjohnson I think there's a factor of 2 missing in the statement here. Commented Mar 24, 2015 at 22:30

This is not exactly an answer, and not exactly giving a condition equivalent to reflexivity, but I want to give a geometric example warning for the development of the geometric intuition.

Let me start from a well-known characterization that a Banach space $X$ is super-reflexive if and only if $X$ can be equivalently renormed with a uniformly convex norm. If you are not familiar with these definitions, please check Wikipedia. Intuitively uniform convexity says that the ball is "uniformly" round. One has that if $x, y$ belong to the unit sphere of $X$ and are $\varepsilon>0$ apart (i.e. $\|x-y\|>\varepsilon$), then the midpoint has to be inside the unit ball, and not on the sphere, and it has to be uniformly "deep", i.e. $$\|\frac{x+y}2\|\le 1- \delta_X(\varepsilon),$$ where $\delta_X(\varepsilon)>0$ and depends only on $\varepsilon$.

Note that the condition is that $X$ can be renormed to satisfy this condition, not that every norm satisfies it. In fact, it is possible to equivalently renorm the Hilbert space $\ell_2$ to have a positive face of the unit sphere of $\ell_1$ (that is a very "flat" set) inside the positive face of renormed $\ell_2$, which is "the most reflexive space". In fact this is possible in every infinite dimensional Banach space.

To see this, let $(x_i, x_i^*)$ in $X\times X^*$ be a biorthogonal system with $\|x_i\| = 1$ and $\|x_i^*\|\leq 2$. (Such a biorthogonal system exists by applying a theorem of Ovsepian and Pelczynski, see for example the book J. Diestel, Sequences and Series in Banach Spaces,page 56, to a separable subspace of $X$ and then extending to functionals on all of $X$ via the Hahn-Banach theorem.)

Then let $$|||x||| = \max\{ |x_1^*(x)|, \frac12 \|x\|, \displaystyle\sup_{i\ne j; \, i, j\geq 2} (\,|x_i^*(x)| + |x_j^*(x)|\, ) \}.$$ This defines an equivalent norm on $X$ with $||| x_1 + x_n|||=1$ and for all $\alpha_n\geq 0$ we have $$|||\, \sum_{n=1}^\infty \alpha_n (x_1 + x_n)\,||| = \sum_{n=1}^\infty \alpha_n.$$

I first learned about this fact from Vitali Milman. I think it goes back to Ptak. The above norm was defined by A. Pelczynski and M. Wojciechowski.

Several geometric properties equivalent to non-reflexivity for a Banach space were studied by R.C. James in "Some self-dual properties of normed linear spaces". Ann. of Math. Studies 69 (1972), 159-175. See also Section 10 of D. van Dulst's book "Reflexive and superreflexive Banach spaces". Math. Centre Tracts 102. Amsterdam 1978. One of them:

A Banach space $X$ is non-reflexive if and only if there exist $\varepsilon>0$ and a sequence $(x_n)$ in the unit ball of $X$ such that for each $k\in N$, $$dist(co\{x_1,\ldots,x_k\}, co\{x_{k+1},\ldots\})\geq \varepsilon.$$