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 <i} langle Ae_i, f_j rowle f_j} { | Ae_i- sum_ {j <i} langle Ae_i, f_j rangle |}.
end {align *}

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.