Let $X$ be a Banach space and $B_X$ be the unit ball.
Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty intersection.
Is it true that $X$ is reflexive?
Let $X$ be a Banach space and $B_X$ be the unit ball.
Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty intersection.
Is it true that $X$ is reflexive?
We will admit the following result:
A Banach space $X$ is reflexive if and only if for all $l:X\rightarrow \mathbb R$ linear and continuous we can find $x_0$ such that $\lVert x_0\rVert =\lVert l\rVert = \sup_{x\neq 0}\frac{l(x)}{\lVert x\rVert}$.
Let $l$ such a map. For all $n\in\mathbb{N}^*$, we can find $x_n$ with $\lVert x_n\rVert =1$ and $\langle l,x_n\rangle \geq \lVert l\rVert -\frac 1n$. Let $C_n$ the closed convex hull of the set $\left\{x_k,k\geq n\right\}$. $\left\{C_n\right\}$ is a decreasing sequence of closed convex non empty subsets of $B_X$. Let $x\in \bigcap_{n\in \mathbb{N}^*}C_n$. Let $\varepsilon>0$ and $k_0\in\mathbb N^*$. We can find $N\in\mathbb N$ and $k_1<\cdots
Such $C_n$ exist in any Banach space. For example take $C_n$ to be the closed ball of radius $1-1/n$ centred at the origin.
It's true This is an old (1939) theorem due to Smulyan Look at any Banach space theory book for the proof