Subtracting the weak limit reduces the norm in the limit

Question

Let $$X$$ be some reflexive Banach space. Suppose $$x_n$$ is some sequence in $$X$$ that weak converges to some $$y \neq 0$$. Is it the case that $$\limsup \|x_n - y\| < \limsup \|x_n\| ?$$

Positive with Hilbert spaces

In the Hilbert space case this is true, as $$\langle x_n - y, x_n -y \rangle = \| x_n\|^2 + \|y\|^2 - \langle x_n, y\rangle - \langle y, x_n\rangle$$ and by weak convergence the latter two terms both converges to $$\|y\|^2$$ and we get $$\limsup \|x_n - y\|^2 = \limsup \|x_n\|^2 - \|y\|^2.$$

A negative example with a non-reflexive Banach space and weak* convergence

Let $$X = L^\infty(\mathbb{R})$$, and take $$x_n = \mathbf{1}_{[-2,-1]} + \mathbf{1}_{[n,n+1]}$$. Then $$x_n$$ weak* converges to $$\mathbf{1}_{[-2,-1]}$$ but $$\|x_n - \mathbf{1}_{[-2,-1]}\|_{\infty} = 1 = \|x_n\|_\infty$$.

• [deleted previous comments which were based on guesswork that the answer below refutes] – Yemon Choi Dec 7 '20 at 18:09

The property you indicate is known as (strict) Opial’s Property (see https://en.m.wikipedia.org/wiki/Opial_property). It fails generally in reflexive spaces; in fact, it fails generally even for uniformly convex spaces where it is equivalent to Delta convergence (see https://en.m.wikipedia.org/wiki/Delta-convergence). Indeed, for $$L^p[0,1]$$, where $$p\ne 2$$, Opial’s Property fails. Using the duality pairing $$\langle\cdot\,,\cdot\rangle\colon X\times X^*\to \mathbb{R}~or~\mathbb{C}$$ and (the normalized) duality map $$J(x):=\{f\in X^*:\|f\|^2=f(x)=\|x\|^2\}\,,$$ Brailey Sims (https://www.mathshunter.edu.au/brailey/Research_papers/As%20Support%20Map%20Characterisation%20of%20the%20Opial%20Conditons.pdf) characterised spaces with the strict Opial Property as those spaces such that whenever $$x_n$$ converges weakly to $$x\ne 0$$, then $$\limsup_{\substack{n\to\infty\\j x_n\in J x_n}}\langle x\,,j x_n\rangle>0\,.$$ (The $$\limsup$$ may be equivalently replaced with $$\liminf$$). From this it is easy to deduce that the sequence spaces $$\ell^p$$, where $$1\le p<\infty$$ have Opial’s Property.