How can one prove that the cardinality of an orthonormal basis of $l^2(A)$ is exactly the cardinality of $A$? I would like to use this result to show that $l^2(A)$ is Hilbert-isomorphic to $l^2(B)$ if and only if $A$ and $B$ have the same cardinality.
Any comment or answer is much appreciated and let me know if I can explain myself more clearly!
In the end I tried to prove it by putting together stuff that I found online (mainly wikipedia, wikiproof and this forum) and by adding some propositions and the concept of Hilbert-dimension. You can find my attempt at the following link (to the pdf): https://drive.google.com/file/d/1N6y_0T2YDb0uB9SQ-9ulIvCmoKZoa_hN/view?usp=drive_link
Let me know if there are errors (both mathematical and grammatical/syntactic).