Given a linear map $l: V^k \rightarrow F$, is there some way to construct a multilinear map that somehow corresponds to this?
Similarly, given a multilinear map $m: V^k \rightarrow F$, is there some linear map that corresponds to this?
I'm trying to build an intuition for "regular" linearity and multilinearity. Any views on how to understand these operations and their relationships would be great.
How do I intuitively think of a multilinear map? Isn't the multi-linearlity property "weird", in that we need all the other coordinates fixed? How do I think about this?
No. You can't require $m$ to be both linear and multilinear.
Pick $v\in V^k$ such that $m(v) = a\ne 0$. Then by linearity we have $m(2v)=2a$ yet by multilinearity we have also $m(2v)=2^ka$.
Multilinearity just means being linear in every component. A very elementary example is the inner product on $\Bbb R^n$ (which is bi-linear), and another common multilinearity map is the $\det$ map from $(\Bbb F^n)^n$ to $\Bbb F$.