# Supporting functional

In convex analysis and mathematical optimization, the **supporting functional** is a generalization of the supporting hyperplane of a set.

## Mathematical definition

Let *X* be a locally convex topological space, and be a convex set, then the continuous linear functional is a supporting functional of *C* at the point if for every .^{[1]}

## Relation to support function

If (where is the dual space of ) is a support function of the set *C*, then if , it follows that defines a supporting functional of *C* at the point such that for any .

## Relation to supporting hyperplane

If is a supporting functional of the convex set *C* at the point such that

then defines a supporting hyperplane to *C* at .^{[2]}