Hahn’s Theorem

The Hahn – Banach Theorem , in functional analysis especially, is a tool used in the problem of the prolongation of a linear functional, a fact that happens frequently. And if that is the case, the following proposition is an instrument of valuable help.

Summary

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• 1 Theorem and corollaries
  • 1 Text of the proposal
  • 2 Motto
  • 3 Corollaries
• 2 Sources
• 3 See also
• 4 External link

Theorem and corollaries

Proposal text

Any bounded linear functional h defined in a linear variety L ₀ of a normed linear space L can be prolonged throughout the space without affecting the growth of the norm of h .

For the clarification of the theorem we present a:

Motto

Any bounded linear functional h defined in a linear array L ₀ , a subset of L and always dense in L, is continuously extended uniquely throughout the space without the norm of h increasing.

In practice his corollaries of the theorem are more used than the same proposition:

Corollaries

• If v is a nonzero element of L, then there is a linear functional hsuch that || h || = 1 and h (v) = || v ||

Let us define in L ₀ = {x: x = tv, t is real} the functional h (x) = t || v || here x = tv. According to the statement, the extension of this functional to the entire L space is the required functional one.

• A support plane to D can be drawn at any point v on a unit disk D
• If L ₀subset of L is a linear variety and v ≠ 0, then there is a linear functional h such that h (x) = 0 for any x of L ₀ , h (v) = 1, in addition

|| h || = 1 ÷ inf || xv || x is at L ₀ .

• two different elements v, w of a locally convex space can always be separated by a linear functional, that is, there is a linear functional h such that

h (v) ≠ h (w)