Functional Analysis – Is Gram-Schmidt Continuous on a Separable Hilbert Space Operator Standard?

Let $$mathcal H$$ to be a separable Hilbert space, with inner product $$langle cdot, cdot rangle$$and orthonormal $$(e_i) _ {i in mathbb N}$$. Consider a continuous linear incorporation $$A colon mathcal H to mathcal H$$. Then we can apply the Gram-Schmidt process to vectors (linearly independent) $$Ae_i$$. That is, we define recursively
begin {align *} f_0 & = Ae_0, \ f_i & = frac {Ae_i- sum_ {j
Define a new operator $$GS (A)$$ by $$Ae_i = f_i$$. Since $$f_i$$ are a family of orthonormal vectors, $$GS (A)$$ will be an isometric integration $$mathcal H to mathcal H$$.

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

Is this card $$GS$$ Continuous compared to topologies of operator standards on domain and target? If the answer is no, what is the largest subspace of $$mathcal E$$ such as the restricted card $$GS$$ is continuous? (obviously $$GS | _ { mathcal I}$$ is the identity)

If that were the case, then especially $$f_i$$ would continually depend on $$A$$ uniformly in $$i$$and I do not even see if that's true.