So in the text Functional Analysis by Walter Rudin the proposition 1.21 states
If $n$ is a positive integer and $Y$ is an $n$-dimensional subspace of a complex topological vector space $X$, then every isomorphism of $C^n$ onto $Y$ is a homeomorphism and $Y$ is a closed set
So can I argue thorugh the above theorem that any linear map between topological vector spaces of finite dimension is a closed map? In particular is any linear transformation $f$ of $\Bbb C^n$ in $Y$ a closed map? So could someone help me, please?
Let be $\pi_1:\Bbb R^2\rightarrow \Bbb R$ the projection onto the first coordinate and so let be $C$ set given by the identity $$ C:=\Big\{(x,y)\in\Bbb R^2:y=\frac 1 x\,\,\,\text{for}\,x>0\Big\} $$ So $C$ is closed, being the graph of a function, but however $\pi_1[C]$ is not closed.