How to show that there is a continuous linear transformation?

Question

Let $H$ be a Hilbert space and $M$ closed subspace of $H$, then following:

Suppose that $M$ has codimension 1, which means that $M^\perp $ have a dimension 1. Then, how to show that there is a continuous linear transformation $ T: H \rightarrow R$ such that $ M=KerT$?

Should we use orthogonal projection theorem and write $<m,V>=0 $ and then defining $KerT$ ?

Answer 1

Let $u$ a non zero element of theorthogonal of $M$, define $T(x)=\langle u,x\rangle$, then $kerT=M$.