Intersection of closed linear spans of a sequence of linearly independent, closed and bounded sets Let $V$ be a Banach space.
Let $\{B_n\}_{n \in \mathbb{N}}$ be a sequence of subsets of $V$ such that $\forall n \in \mathbb{N}:$
$B_n$ is linearly independent, closed and bounded, $B_{n+1} \subsetneq B_n$ and
$\bigcap_{n=1}^\infty B_n = \{b_0\}$ with $b_0 \in V$
Is it true that
$$
\bigcap_{n=1}^\infty
\overline{ 
\operatorname{span}
B_n
}
=
\operatorname{span}
\{b_0\}
$$
 A: No. $V=L^2([1,2])$ and $$f_n(x) = \sqrt{\frac{2n+1}{2^{2n+1}-1}}\,x^n, \qquad n=0,1,2,\dots$$ The functions $f_n$ have unit $L^2$ norm and are linearly independent. Also, the set $\{f_n\}$ has no limit points because any subsequence $\{f_{n_k}\}$ converges to $0$ uniformly on any interval $[1,2-\epsilon]$ and thus cannot converge to a nonzero element of $L^2([1,2])$ in the $L^2$ norm. 
Claim: For each $n\in \mathbb{N}$ the set $F_n = \{f_k:k\ge n\}$ has dense span in $L^2([0,1])$. 
Proof: Every function $f\in L^2([1,2])$ can be approximated by a continuous function $g\in C([1,2])$. The function $h(x) = g(x)/x^n$ is also continuous on $[1,2]$ and therefore is a uniform limit of some polynomials $\{P_j:j\in\mathbb{N}\}$. The sequence of polynomials $\{x^n P_j:j\in\mathbb{N}\}$ is in the linear span of $F_n$ and converges to $g$ uniformly, hence in $L^2$. $\quad\Box$ 
Let $ B_n = \{f_0\}\cup \{f_k:k\ge n\}$. The sets $B_n$ satisfy all requirements, and $\bigcap_{n=1}^\infty = \{f_0\}$ but 
$$\bigcap_{n=1}^\infty \overline{ \operatorname{span} B_n } = V$$ because $\overline{ \operatorname{span} B_n }  = V$ for every $n$.
Aside
Such counterexamples exist because "linearly independent set with dense linear span" is quite far from what we think of a  "basis". Such a set can be very redundant: throwing out many (even infinitely many) elements, we may still have dense linear span. The root cause is that the purely algebraic notion of linear independence is not of much help when topology comes into play. The definition of a Schauder basis imposes a stronger notion of independence, where the relation $\sum c_n b_n = 0$ implies $\forall n\ c_n=0$ even for infinite sums.
