Suppose $A$ is a closed subset of a smooth manifold $M$. Denote by $C(M)$ continuous functions with uniform norm. When is the subspace of functions vanishing on $A$ complemented in $C(M)$?
This is true if $A$ is a finite set of points (since then it's finite codimensional and closed) and if $A$ is a retraction of $M$.
An easier question I don't know how to answer is whether continuous functions on the disk that vanish on the boundary are complemented in all continuous functions on the disk.
Thanks in advance!
Best Answer
After doing some research, I found out that the answer is always yes: the subspace $S$ of functions vanishing on $A$ is always complemented in $C(M)$, whenever $M$ is any compact metric space and $A$ is any closed subset.
To prove this, it is enough to show that there exists a family $\{\mu_x\}_{x\in M}$ of finite signed measures on $A$ with the following two properties:
(i) $x \mapsto \mu_x$ is weakly continuous.
(ii) $\mu_x=\delta_x$ for every $x \in A$, where $\delta_x$ is a point mass at $x$.
Indeed, if such a family of measures exists, then we can define a map $T:C(A) \to C(M)$ by defining $Tf(x):=\int_A f(y)\mu_x(dy)$. Then the image of $T$ is easily checked to be a complement for $S$. Indeed, the image is closed because $\|Tf\|_{C(M)} \geq \|f\|_{C(A)}$ trivially. It trivially intersects $S$ because any $g=Tf \in S \cap T(C(A))$ satisfies $$f(x)=\int_A f(y) \delta_x(dy)=Tf(x)=g(x)=0 \;\;\;\;for \;\;\;\;x \in A.$$ And finally, $S+T(C(A))=C(M)$ because we can write any function $f\in C(M)$ as $(f-\phi)+\phi$ where $\phi = T(f|_A)$. Then clearly $f-\phi\in S$ and $\phi \in T(C(A))$.
So let's prove that such a family of measures always exists for any closed subset $A$ of a compact metric space $M$. We are going to use a generalization of the Tietze extension theorem which holds whenever the target space is a locally convex topological vector space.
Let $E$ be the space of all finite signed measures on $A$, i.e., $E=C(A)^*$. Equip $E$ with the weak topology, i.e., the topology generated by the family of seminorms $N_f(\mu) = |\mu(f)|$, with $f\in C(A)$.
Define a map $u:A \to E$ by sending $x \mapsto \delta_x$. Obviously this is continuous. Therefore by the Tietze extension theorem, $u$ can be extended continuously to a map $v: M \to E$. Then set $\mu_x:=v(x)$.