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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.

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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.

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