Hahn–Banach theorem

The Hahn- Banach (after Hans Hahn and Stefan Banach ) from the mathematical branch of functional analysis is one of the starting points of the functional analysis. He ensures the existence of sufficiently many continuous linear functionals on normed vector spaces, or more generally, on locally convex spaces. The investigation of a space with the help of the defined thereon steady, linear functionals leads to a far-reaching duality theory, which is not possible in general topological vector spaces in this form, since the Hahn- Banach analogous statement does not apply there.

In addition, the Hahn- Banach theorem is the basis for many non- constructive existence proofs, such as the separation theorem or the Krein - Milman.

The finite-dimensional case

If one vectors of a finite-dimensional real or complex vector space with respect to a fixed basis chosen in the form of a row vector is, so you can see the respective th entries of this row vectors as functions

Understand (this is the basic body or ). An essential part of the meaning of such a well-known from linear algebra coordinate representation lies in the fact that two vectors are equal if and only if all of its coordinates are the same:

Therefore, the coordinate functions separate the points, that is, are different vectors, then there is an index, then that. They are continuous linear functionals on the space coordinates.

In infinite-dimensional spaces, there are usually not the coordinates of similar design, if you insist on doing continuity of the coordinates. The Hahn- Banach implies, however, that the set of all continuous linear functionals on a normed space (or more generally on a locally convex space ) separates the points.

Formulation

It is a vector space over.

Be it

• A linear subspace;
• A sub-linear mapping;
• A linear functional, then for the for all.

Then there is a linear functional such that

• And

Applies to all.

The proof of this fundamental theorem is not constructive. We consider the set of all sequels with on subspaces applies for all. Then you point the lemma of Zorn that the set of all such sequels has maximal elements and that such a maximal element is a sought-after sequel.

Corollaries

Often one of the following statements, which can be derived from the above set easily, meant when the " Hahn- Banach " is quoted:

• If a normed space, so there is for each a linear functional with norm 1, then for the. Are different points, we obtain the above-mentioned property of the separation points, by applying this on.
• More generally, if a normed space, a subspace, and not in the conclusion of, so there is a linear functional with norm 1, which applies to disappear and for the.
• If a normed space, a subspace and a continuous linear functional on, it can be continued to a continuous linear functional on all of the same standard. In other words, the limitation of a functional onto mapping of the dual- space.
• Further implications of geometric nature can be found in Article separation theorem.