We have the following theorem
Let $T:X\rightarrow Y$ be a compact operator between the Hilbert spaces $X,Y$. Then there exist (possibly finite) orthonormal bases $\{e_1,e_2,\ldots \}$ and $\{f_1,f_2,\ldots \}$ of $X,Y$ and (possibly finite) numbers $s_1,s_2,\ldots$ such that $s_n \rightarrow 0$ (if there are countably many numbers), such that $ Tx=\sum\limits_{n=1}^{\infty}s_n \left
f_n $
This is the singular value decomposition for operators between Hilbert spaces.
My questions are: If $X$ and $Y$ are finite dimensional then the theorem, as stated above is true? If $X$ is finite dimensional, both orthonormal bases become finite, since then we work in $T(X)$ which is a finite dimensional subspace in the (possible infinite dimensional) space $Y$.
But what happens, if $X$ is infinite dimensional, but $Y$ is not? Are the orthonormal bases in that case still finite?