For what matrices are generalized inverses A+ of matrices defined? For only Finite dimensional or both finite and infinite dim'l, i.e. dimension does not matter?

If generalized inverses are defined for infinite dim'l matrices, then I have a trouble in Penrose's proof for the existence of generalized inverses of matrices in 1955: (Penrose,1955) (A*A), (A*A)^2, (A*A)^3,... CANNOT BE LINEARLY INDEPENDENT.

Trouble: For finite dimensional mxn matrices, I okey (A*A), (A*A)^2, (A*A)^3,... are linearly dependent since the dimension of vectore space of all mxn matrices are mn. This is a stopping reason for that sequence.

But, how can we say that (A*A), (A*A)^2, (A*A)^3,... are linearly dependent in the infinite dimensional case as well? I cannot see any stopping reason here. I looked Penrose's proof (of 1955), but there, Penrose did not write any restrictive words about dim of matrices to only finite dimensional case.

Note: I attached the Penrose's 1955 original proof as well.
The dimension for Generalized Inverses of matrices and troubled Penrose's proof-p1.jpgThe dimension for Generalized Inverses of matrices and troubled Penrose's proof-p2.jpgThe dimension for Generalized Inverses of matrices and troubled Penrose's proof-p3.jpg