On a property for normed spaces

Question

I came upon the following specific property for a normed space $X$, and I am looking for a characterization of the normed spaces where it holds true:

If a sequence $x_n$ in $X$ satisfies $\displaystyle \lim_{n\to\infty}(\|x_n+y\|-\|x_n\|)=\|y\|$ for all $y\in X$, then $\displaystyle \lim_{n\to\infty}x_n=0$.

This is not true in $l_1$; take $x_n=e_n$, the unit vector basis. The same counterexample doesn't seem to work in $c_0$, $l_\infty$, and $l_p$ for $p\neq 1$. Is this actually true in these spaces, or is this property, in fact, never satisfied?

Answer 1

Let me give you a full, but not satisfactory, general characterization of your property, and then a much nicer characterization of your property among separable spaces.

First we say that a sequence $(x_n)$ is $L$-orthogonal if $\|x_n + x \| \rightarrow 1 + \|x\|$ for every $x \in X$; and we say that an element $x^{**} \in X^{**}$ is $L$-orthogonal if $\|x^{**} + x\| = \|x^*\| + \|x\|$ for every $x \in X$.

Proposition. A Banach space $X$ has your property iff it does not contain an $L$-orthogonal sequence.

Proof. First note that any $L$-orthogonal sequence $(x_n)$ satisfies the assumption of your property, but $\|x_n\| \rightarrow 1$, so spaces containing $L$-orthogonal sequences cannot have your property. On the other hand suppose that $X$ does not satisfy your property, witnessed by a sequence $(x_n)$. As $(x_n)$ is bounded and not converging to zero, there is $c > 0$ and a subsequence $(y_n)$ of the sequence $(x_n)$ such that $\|y_n\| \rightarrow c$. But then $z_n = c^{-1} y_n$ forms an $L$-orthogonal sequence. QED

$L$-orthogonal sequences have been studied, so you can find something about your spaces. Let me mention some known results (for them and more see this paper by Avilés et. al. and references therein).

1. A Banach space $X$ contains $\ell_1$ iff there is an $L$-orthogonal element in $X^{**}$.
2. If $X$ is a Banach space of density $\leq \mathfrak{p}$ (the pseudointersection number) and $X$ contains $L$-orthogonal sequence, then there is an $L$-orthogonal element in $X^{**}$. This particulary holds for separable $X$.
3. If $X$ is separable Banach space and $X^{**}$ contains $L$-orthogonal element, then $X$ contains an $L$-orthogonal sequence.

In particular, combining the results above, you get:

Proposition. Let $X$ be a separable Banach space. Then $X$ has your property iff $X$ does not contain $\ell_1$.

The proposition above gives you that all the spaces $c_0,\ell_p$, $1 < p < \infty$ have your property.