All vector spaces are assumed to be real and finite dimensional.
I am trying to show that:
Let $V$ be a vector space and $f:\bigoplus^k V\to \bigwedge^k V$ be defined as $$f(v_1, \ldots, v_k)= v_1\wedge \cdots\wedge v_k$$ Then $f$ is smooth.
In general, I am looking for a criterion for smoothness of a map $f:M\to \bigwedge^k V$ from a smooth manifold $M$ into the exterior power of a vector space.
For example, a map $f:M\to V$ is smooth if and only if $\omega\circ f:M\to \mathbf R$ is smooth for all $\omega\in V^*$.
Thanks.