The title says it all:

Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The strong an the weak one coincide, since $M$ is compact)

Hirsch's book on differential topology states without proof that this space is completely metrizable, second countable and locally contractible.

However, is it locally compact?