Dual space

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. The dual space of $V$ is the vector space $$V^* = \Hom_F(V, F)$$ of all linear forms on $V$. Its operations are pointwise: $(f + g)(v) = f(v) + g(v)$ and $(af)(v) = a f(v)$.

Remarks

It is a special case of the space of linear maps. If $\dim V = n$ is finite, then $\dim V^* = n$, see the dimension of the dual space. The elements of $F^n$ act on column vectors by the dot product, and this identifies $(F^n)^*$ with $F^n$. For infinite-dimensional $V$, the space $V^*$ is not isomorphic to $V$.

Explanations

L1 human

The dual space collects all the ways to measure a vector by one linear number. On $\R^3$, each linear form is a dot product with a fixed vector. So, in finite dimensions, the dual space looks like the original space, but its elements play a different role.