HAHN–BANACH THEOREM

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In mathematics, the 'Hahn–Banach theorem' is a central tool in functional analysis. It allows the extension of bounded linear operators defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space interesting. It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the 1920s.

Contents

Formulation

Important consequences

Hahn-Banach separation theorem

Relation to the axiom of choice

See also

References

Notes

Formulation

The most general formulation of the theorem needs some preparation. Given a vector space ''V'' over the scalar field 'K' (either the real numbers 'R' or the complex numbers 'C'), a function ''N'' : ''V'' → 'R' is called ''sublinear'' if
:''N''(''ax'' + ''by'') ≤ |''a''| ''N''(''x'') + |b| ''N''(''y'')
for all ''x'' and ''y'' in ''V'' and all scalars ''a'' and ''b'' in 'K'. Every norm on ''V'' is sublinear, as is every seminorm, but there are other examples.
The Hahn–Banach theorem states the following:
: Let ''N'' : ''V'' → 'R' be sublinear, let φ : ''U'' → 'K' be a linear functional on a subspace ''U'' of ''V''. If φ is dominated by ''N'' on ''U'' (meaning |φ(''x'')| ≤ ''N''(''x'') for all ''x'' in ''U'') then there exists a linear extension ψ : ''V'' → 'K' of φ to all of ''V'' (meaning ψ(''x'') = φ(''x'') for all ''x'' in ''U'') which is dominated by ''N'' on all of ''V''.
The extension ψ is in general not uniquely specified by φ and the proof gives no method as to how to find ψ: in the case of an infinite dimensional space ''V'', it depends on Zorn's lemma, one formulation of the axiom of choice.
In fact, the sublinearity condition on ''N'' can be slightly relaxed: it suffices to assume that
:''N''(''ax'' + ''by'') ≤ |''a''| ''N''(''x'') + |''b''| ''N''(''y'')
for all ''a'' and ''b'' in 'K' with |''a''| + |''b''| = 1 (Reed and Simon, 1980).
The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.

Important consequences

The theorem has several important consequences, some of are which also sometimes called "Hahn–Banach theorem":

★ If ''V'' is a normed vector space with subspace ''U'' (not necessarily closed) and if φ : ''U'' → 'K' is continuous and linear, then there exists an extension ψ : ''V'' → 'K' of φ which is also continuous and linear and which has the same norm as φ (see Banach space for a discussion of the norm of a linear map).

★ If ''V'' is a normed vector space with subspace ''U'' (not necessarily closed) and if ''z'' is an element of ''V'' not in the closure of ''U'', then there exists a continuous linear map ψ : ''V'' → 'K' with ψ(''x'') = 0 for all ''x'' in ''U'', ψ(''z'') = 1, and ||ψ|| = ||''z''||⁻¹.

Hahn-Banach separation theorem

Another version of Hahn-Banach theorem is known as 'Hahn-Banach separation theorem'.^[1][2] It has numerous uses in complex geometry.^[3]
'Theorem:' Let ''V'' be a topological vector space over $\{Bbb\; K\}=\{Bbb\; R\}$ or $\{Bbb\; C\}$, and ''A'', ''B'' convex, non-empty subsets of ''V''. Assume that $Acap\; B=emptyset$. Then
(i) If ''A'' is open, there exists a continuous linear map $lambda:;\; Vmapsto\; \{Bbb\; K\}$ and $tin\; \{Bbb\; R\}$ such that
:
for all $ain\; A,\; b\; in\; B$
(ii) If ''V'' is locally convex, ''A'' is compact, and ''B'' closed, there exists a continuous linear map $lambda:;\; Vmapsto\; \{Bbb\; K\}$ and $tin\; \{Bbb\; R\}$ such that
:
for all $ain\; A,\; b\; in\; B$.

Relation to the axiom of choice

As mentioned earlier, the axiom of choice implies the Hahn-Banach theorem. The converse is not true. One way to see that by noting that the ultrafilter lemma, which is strictly weaker than the axiom of choice, can be used to show the Hahn-Banach theorem, although the converse is not the case.

See also

★ M. Riesz extension theorem

References

★ Lawrence Narici and Edward Beckenstein, "The Hahn–Banach Theorem: The Life and Times", ''Topology and its Applications'', Volume '77', Issue 2 (1997) Pages 193-211.

★ Michael Reed and Barry Simon, ''Functional Analysis,'' Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.

Notes

1. Klaus Thomsen, The Hahn-Banach separation theorem, Aarhus University, Advanced Analysis lecture notes
2. Gabriel Nagy, Real Analysis lecture notes
3. R. Harvey and H. B. Lawson, "An intrinsic characterisation of Kahler manifolds," Invent. Math 74 (1983) 169-198.