Definition of the problem
Let $\mathcal{H}$ be a separable Hilbert space on $J\subset\mathbb{N}$ an index set. Let $\Phi:=\left(\varphi_{j}\right)_{j\in J}\subset\mathcal{H}$ be a frame for $\mathcal{H}$.
I have to prove that the analysis operator $T_{\Phi}:\mathcal{H}\rightarrow\ell_{2}\left(J\right)$ of the frame $\Phi$, defined by $ T_{\Phi}x:=\left(\left\langle x,\varphi_{j}\right\rangle \right)_{j\in J},\quad x\in\mathcal{H}, $ is injective and has a closed range.
Effort to prove closed range
We need to show that $ ran\, T_{\Phi}closed\Leftrightarrow\forall x_{n}\in\mathcal{H}:\,\lim_{n\rightarrow\infty}T_{\Phi}x_{n}\in\ell_{2}\left(J\right). $
Let $x_{n}\in\mathcal{H}$. We have that $ \lim_{n\rightarrow\infty}T_{\Phi}x_{n}=\lim_{n\rightarrow\infty}\left(\left\langle x_{n},\varphi_{j}\right\rangle \right)_{j\in J}\overset{?}{\in}\ell_{2}\left(J\right). $
For this statement to hold, we have a look at $ \sum_{j\in J}\left|\lim_{n\rightarrow\infty}\left\langle x_{n},\varphi_{j}\right\rangle \right|^{2}. $
Using the fact that $\Phi$is a frame for $\mathcal{H}$, $ \sum_{j\in J}\left|\lim_{n\rightarrow\infty}\left\langle x_{n},\varphi_{j}\right\rangle \right|^{2}\overset{???}{=}\sum_{j\in J}\lim_{n\rightarrow\infty}\left|\left\langle x_{n},\varphi_{j}\right\rangle \right|^{2}=\lim_{n\rightarrow\infty}\sum_{j\in J}\left|\left\langle x_{n},\varphi_{j}\right\rangle \right|^{2}\leq\lim_{n\rightarrow\infty}B\left\Vert x_{n}\right\Vert ^{2}. $
My question 1
How could I use that to show that $T_{\Phi}$has a closed range? Would an upper bound help me at all with this?
Effort to show that $T_{\Phi}$is injective
Denote $\left\{ e_{i}:i\in I\right\} $be an orthonormal basis. Let $x,y\in\mathcal{H}$. Assume $T_{\Phi}x=T_{\Phi}y$. We have that $ \sum_{i\in J}\left\langle x,\varphi_{i}\right\rangle e_{i}=\sum_{i\in J}\left\langle y,\varphi_{i}\right\rangle e_{i} $ $ \Leftrightarrow\forall i\in J:\quad\left\langle x,\varphi_{i}\right\rangle =\left\langle y,\varphi_{i}\right\rangle . $
My question 2
Am I allowed to make the assumption on the orthornormal basis? How can I show that such an orthonormal basis exists? How could I go any further showing that it is injective? We do not know if the inner product $\left\langle \cdot,\cdot\right\rangle $ is one-to-one?!
Thank you, Franck!