In a sense, linear functionals are co-probes for vector spaces. If the vector space $V$ in question has finite dimension and is equipped with a basis, then all linear functionals are linear combinations of the coordinate projections. These projections comprise the dual basis.

In infinite-dimensional topological vector spaces, the notion of dual basis breaks down once spaces more general than Hilbert spaces are considered. But for locally convex spaces, the Hahn–Banach theorem ensures the existence of ‘enough’ continuous linear functionals. Among non-LCSes, however, there are examples such that the only continuous linear functional is the constant map onto $0 \in k$.