I am studying topological vector spaces from Sevres' book "Topological vector spaces, distributions and Kernels". In one of the preparatory chapters I encountered the following excercise:
Consider a linear subspace M of a vector space E, and a linear map T from M into F. Prove the existence of an extension T': E into F. (Obviously all the spaces may be infinite dimensional)
I have studied a bit of functional analysis so I think that maybe one could mimic the proof of Hahn-Banach theorem and in this context one should neither take care of the continuity of the extension. However I recall that this proof uses explicitly Zorn Lemma, or the Axiom of Choice. Being a bachelor's student, I was wondering if in this more simple context this could be avoided. I was trying to exploit quotient spaces or to find a decomposition by means of direct sum E= M + M' but was not successful. Any idea ?
Consider the case $F=M$ and $T$ the identity map on $M$. An extension of this to $T': E \to M$ is a projection of $E$ on $M$. Then with $N = \ker(T')$ you can write $E = M \oplus N$.
According to Asaf Karagila's answer here, the Axiom of Choice is equivalent to the statement that this can always be done.