Let $\mathcal H$ be a separable Hilbert space, with inner product $\langle\cdot,\cdot\rangle$, and with orthonormal basis $(e_i)_{i\in\mathbb N}$. Consider a continuous linear embedding $A\colon\mathcal H\to\mathcal H$. Then one can apply the Gram-Schmidt process to the (linearly independent) vectors $Ae_i$. That is, we define recursively $$ f_i=\frac{Ae_i-\sum_{j<i}\langle Ae_i,f_j\rangle f_j}{\|Ae_i-\sum_{j<i}\langle Ae_i,f_j\rangle\|}. $$ Define a new operator $GS(A)$ by $Ae_i=f_i$. Since the $f_i$ are an orthonormal family of vectors, $GS(A)$ will be an isometric embedding $\mathcal H\to\mathcal H$.

Let $\mathcal E$ be the space of all continuous linear embeddings $\mathcal H\to\mathcal H$, and $\mathcal I$ the space of all isometric embeddings $\mathcal H\to\mathcal H$. Then the above construction defines a map $GS\colon\mathcal E\to\mathcal I$.

Is this map $GS$ continuous with respect to the operator norm topologies on both domain and target? If the answer is no, what is the largest subspace of $\mathcal E$ such that the restricted map $GS$ is continuous? (obviously $GS|_{\mathcal I}$ is the identity) Also if the answer is no: Is there a continuous alternative, i.e. a continuous retraction $\mathcal E\to\mathcal I$?

If it were, then the in particular the $f_i$ would depend continuously on $A$ *uniformly in $i$*, and I do not even see if this is true.

injective bounded linear operator with closed range"? $\endgroup$