Let $K$ be a subset of the positive integers $\mathbb{N}$. For each $n\in \mathbb{N}$, $K_{n}$ denotes the set $\{k\in K: k\leq n\}$ and $|K_{n}|$ denotes the number of the elements in $K_{n}$. The natural density of $K$ is defined by $$\delta(K)=\lim_{n\rightarrow \infty}\frac{|K_{n}|}{n}.$$ A sequence $(x_{k})_{k}$ in a Banach space $X$ is said to be norm statistically convergent to $x\in X$ if $\delta(\{k\in \mathbb{N}:\|x_{k}-x\|\geq \epsilon\})=0$ for every $\epsilon>0$.

It is natural to define a series $\sum_{k=1}^{\infty}x_{k}$ to be norm statistically convergent to $x\in X$ by $(s_{n})_{n}=(\sum_{k=1}^{n}x_{k})_{n}$ to be norm statistically convergent to $x$.

We say that a sequence $(x_{k})_{k}$ in a Banach space $X$ is a statistical basis for $X$ if, for each $x\in X$, there exists a unique $(a_{k})_{k}$ such that the series $\sum_{k=1}^{\infty}a_{k}x_{k}$ norm statistically convergent to $x$.

Clearly, statistical basis is a generalization of the notion of Schauder basis.

Question: For each $n$, we can define a linear functional $f_{n}$ in $X$ by $f_{n}(x)=a_{n}$. If $(x_{k})_{k}$ is a Schauder basis, we know that $f_{n}$ is continuous. However, if $(x_{k})_{k}$ is a statistical basis for $X$, is $f_{n}$ continuous?