Let \(M\) be a complex manifold (\(\dim M = n <\infty\)) and fix \(k = 0, 1,2,\dots,\infty,\omega\). Maps of the circle \(S^ 1\) into \(M\), of class \(C^ k\), constitute the \(C^ k\) loop space \(X\) of \(M\). It is known that \(X\) carries the structure of an infinite dimensional complex manifold, modeled on the locally convex topological vector space of \(C^ k\) maps \(S^ 1\to \mathbb C^ n\). Let \(Y=\{x\in X;\;x_{| A}=x_{0| A}\}\) be a submanifold of \(X\), where \(x_0\in X\) and \(A\subset S^ 1\) is a closed subset of \(S^ 1\). The space \(Y\) can be endowed with the structure of an infinite dimensional complex manifold. In this paper, the author identifies the space of holomorphic functions on \(Y\), in particular, the complete description of the space \({\mathcal O}(Y)\) of holomorphic functions on \(Y\), when the loop space is formed for the projective space \(M=\mathbb P_ n\), is given. For example, if \(A\) is the empty set, then the family of compact complex submanifolds of \(Y=X\) is large enough to ensure that all \(f\in \mathcal O(Y)\) are constant.