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Let $(\varphi_i)_{i \in I}$ be a sequence in $\mathcal H$. I want to show $(i) \Rightarrow (ii)$, where

$(i)$ The synthesis operator $T^{*}c = \sum_{i \in I} c_i x_i$ is well-defined for each $c = (c_i)_{i \in I} \in \ell_2(I)$ and maps $\ell_2(I)$ bijectively onto $\mathcal H$.

$(ii)$ There exists an orthonormal basis $(\psi_i)_{i \in I}$ for $\mathcal H$ and a bounded bijective operator $Q \in L(\mathcal H)$ such that $Q\psi_i = \varphi_i$ for every $i \in I$.

It's easy to see that if (i) holds, $(\varphi_i)_{i \in I} = (T^*e_i)_{i \in I}$, where $(e_i)_{i \in I}$ is the canonical base of $\ell_2(I)$. My idea is to use that if we have $(\kappa_i)_{i \in I}$ an orthonormal basis for $\mathcal H$, then $\mathcal H$ is isometrically isomorph to $\ell_2(I)$. The problem is that $\mathcal H$ might not be seperable and especially $I$ not countable, so I don't even know whether there exists an orthonormal basis with this cardinality.

Note that we don't use further equivalences such as $(ii)$ is equivalent to $(\varphi_i)_{i \in I}$ being a Riesz basis.

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By the open mapping theorem $T^*$ is an invertible linear map between $\ell^2(I)$ and $\mathcal H$. The existence of such a map implies that any orthonormal basis of $\mathcal H$ has cardinality $|I|$: the proof given here works. I reproduce its sketch: take each element of the ONB of one space $, send it by that linear map to the second space, expand in its basis (using countably many basis elements of the second space). In this process you will need all the basis elements of the second space; otherwise the linear map would not be onto. Since multiplication by a countable set does not increase the cardinality of an infinite set, it follows that the ONB of the second space does not have greater cardinality than the ONB of the first.

Thus, there is an ONB $(\psi_i)_{i\in I}$ of $\mathcal H$. The map $S:e_i\mapsto \psi_i$, extended by linearity, is an isomorphic isomorphism between $\ell^2(I)$ and $\mathcal H$. Finally, the composition $S(T^*)^{-1}$ is a bounded bijective operator on $\mathcal H$ that maps $\psi_i$ to $\varphi_i$.

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