Characterisation of norm convergence

Question

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$):

1. We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and
2. we have $x_n \xrightarrow{\sigma} x$ (weak convergence) as well as $\|x_n\| \to \|x\|$.

Proof: Just expand the expression $\|x_n - x\|^2 = (x_n - x, x_n - x)$.

Q1: This is not true in arbitrary Banach spaces (correct?). Is there a nice example?

Q2: Does the above characterisation still hold for certain spaces which are not (pre-)Hilbertian? Does it still hold for $L^p$ with $p \ne 2$ e.g.?

edit: Apparently, this is referred to as the Radon–Riesz property, Kadets–Klee property or property (H).

Answer 1

for Q2 - in general it holds for uniformly convex spaces

Answer 2

For Q1: Here's an example in the Banach space $c_0$ of sequences of (real or complex) numbers converging to $0$ with sup norm. Let $x_n=(1,0,0,\ldots,0,0,1,0,0,\ldots)$, with $1$s in the first and $n^\text{th}$ spots and $0$s elsewhere. Then $(x_n)$ converges weakly to $x=(1,0,0,\ldots)$, and $\|x_n\|=\|x\|=1$ for all $n$, but $x_n$ does not converge in norm.