I am looking to prove some parts of Bari's theorem (1946). Let $H$ be a Hilbert space, and let $(e\_n)\_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. Let $(x\_n)\_{n \in \mathbb{N}}$ be a sequence of normalized vectors in $H$ which we assume to be minimal, in the sense that 
\begin{equation}
\forall n \in \mathbb{N}, \quad \text{dist}(x\_n, \overline{\text{span}\{x\_k, \ k \in \mathbb{N}, k \neq n \}}) > 0.
\end{equation}
Moreover, suppose that 
\begin{equation}
\|e\_n - x\_n \|\_H \leq d\_n \quad \forall n \in \mathbb{N},
\end{equation}
where $d\_n$ are nonnegative numbers in $\ell^2(\mathbb{N})$. I am looking to prove the following assertions. First



> 
> \begin{equation}\text{codim}(\overline{\text{span} \{x\_k, k > N \}}) = N
> \end{equation}
> 
> 
> 


where $N$ is a fixed given integer and codim is the codimension of $H\_0:=\overline{\text{span} \{x\_k, k > N \}}$ in $H$.


Next, having proven that the sequence $(y\_k)\_{k \in \mathbb{N}}$ given by $y\_k = e\_k$ for $k \leq N$ and $y\_k = x\_k$ for $k > N$ is a Riesz basis in $H$, that $(x\_k)\_{k > N}$ is a Riesz basis in $H\_0$ and that 
\begin{equation}
H = H\_0 \oplus K,
\end{equation}


where $K = \overline{\text{span}\{x\_k, k \leq N \}}$, I wanted to conclude by showing that



> 
> \begin{equation}
> (x\_k)\_{k \in \mathbb{N}} \text{ is a Riesz basis in } H,
> \end{equation} 
> 
> 
> 


but I failed in doing so. 



On some points of Bari's theorem
By the assumption $\sum\_{n} \|e\_n-x\_n\|^2<+\infty$, the map $e\_n\mapsto e\_n-x\_n$ extends uniquely to a bounded linear operator $A$ on $H$. In fact, $A$ is Hilbert-Schmidt with $\|A\|\_{HS}^2=\sum\_{n} \|e\_n-x\_n\|^2$, in particular compact. Also, the operator $I-A$ (mapping $e\_n$ to $x\_n$) is injective (the existence of a non-zero element $v=\sum\_n v\_ne\_n$ in $\ker(I-A)$ would contradict the minimality of the sequence $\{x\_k\}\_k$). By the Fredholm alternative, $I-A$ is invertible, whence both statements in the boxes follow.


