Question

(Eigenfunction expansion)

If you have a differential equation such as:\[\frac{∂u}{∂t}=\alpha^2 \frac{∂^2u}{∂x^2}+f(x,t)\]\[u(0,t)=0\]\[u(1,t)=0\]\[u(x,0)=g(x)\]
Of which you are solving via eigenfunction expansion;

Why can you assume that f(x,t) is decomposable into:
\[f(x,t)=f_1(t)X_1(x)+f_2(t)X_2(x)+...+f_n(t)X_n(x)\]