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A linear functional dominated by a sublinear function extends to the whole space.
Hahn–Banach (real version): Let be a real vector space, let be a linear subspace, and let be sublinear.
If is a linear functional satisfying
then there exists a linear functional such that
The theorem is a principal tool for convex separation results, such as geometric Hahn–Banach separation.
A vector space over a field is a set with operations (functions) and , and an element , satisfying the following for all and :
Let be a vector space over a field , and let . The set is a linear subspace of if
With the inherited operations, is itself a vector space.
Let be a vector space and let .
Sublinear functions appear as domination bounds in the real Hahn–Banach theorem. A primary source of sublinear functions is the Minkowski gauge of an absorbing convex set.
Let and be vector spaces over the same field .
A function is a linear operator (or linear transformation) if:
Equivalently, is linear if and only if
One often writes instead of .
Linear operators are the morphisms of vector spaces. Their kernels and images give canonical subspaces, and the associated quotient describes the operator up to isomorphism.
Let be a real vector space, let , and let be convex. Assume that core(Ω) is nonempty and .
Theorem. The sets and can be separated by a hyperplane. Equivalently, there exists a nonzero linear functional such that