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| In [[mathematics]], the '''Hahn–Banach theorem''' is a central tool in [[functional analysis]]. It allows the extension of [[Bounded operator|bounded linear functionals]] defined on a subspace of some [[vector space]] to the whole space, and it also shows that there are "enough" [[continuous function (topology)|continuous]] linear functionals defined on every [[normed vector space]] to make the study of the [[dual space]] "interesting". Another version of Hahn–Banach theorem is known as '''Hahn–Banach separation theorem''' or the [[separating hyperplane theorem]], and has numerous uses in [[convex geometry]]. It is named for [[Hans Hahn (mathematician)|Hans Hahn]] and [[Stefan Banach]] who proved this theorem independently in the late 1920s, although a special case <ref>for the space ''C''[''a'', ''b''] of continuous functions on an interval</ref> was proved earlier (in 1912) by [[Eduard Helly]],<ref>{{MacTutor Biography|id=Helly}}</ref> and a general extension theorem from which the Hahn–Banach theorem can be derived was proved in 1923 by [[Marcel Riesz]].<ref>See [[M. Riesz extension theorem]]. According to {{cite journal|mr=0256837|last=Gȧrding|first=L.|author-link=Lars Gårding|title=Marcel Riesz in memoriam|journal=Acta Math.|volume=124|year=1970|issue=1|pages=I–XI|ref=harv}}, the argument was known to Riesz already in 1918.</ref>
| | If you have the desire to procedure settings immediately, loading files fast, but the body is logy plus torpid, what would you do? If you are a giant "switchboard" that is deficiency of efficient management system and effective housekeeper, what would we do? If you have send the exact commands to a notice, yet the body cannot perform correctly, what would we do? Yes! You want a full-featured repair registry!<br><br>If it happens to be not because big of the issue as we think it happens to be, it will probably be solved conveniently by running a Startup Repair or by System Restore Utility. Again it may be because effortless because running an anti-virus check or cleaning the registry.<br><br>It doesn't matter whether you're not especially obvious about what rundll32.exe is. But remember which it plays an significant part inside maintaining the stability of the computers plus the integrity of the program. When several software or hardware could not answer normally to a program operation, comes the rundll32 exe error, which will be caused by corrupted files or missing information inside registry. Usually, error content may shows up at booting or the beginning of running a system.<br><br>If you feel we don't have enough money at the time to upgrade, then the greatest choice is to free up some room by deleting a few of the unwelcome files and folders.<br><br>There are numerous [http://bestregistrycleanerfix.com/registry-reviver registry reviver] s found on the market now. How do you understand that you to choose? Well, when you bought a car you did several research on it, didn't you? You didn't only go out plus purchase the initial red convertible you saw. The same thing works with registry cleaners. On any search engine, type in "registry cleaner reviews" and they may receive posted for you to read about.<br><br>The initial thing you need to do is to reinstall any system which shows the error. It's typical for various computers to have particular programs which require this DLL to show the error when we try plus load it up. If you see a particular program show the error, you need to first uninstall that system, restart a PC plus then resinstall the system again. This must replace the damaged ac1st16.dll file and remedy the error.<br><br>We need an option to automatically delete unwanted registry keys. This will protect you hours of laborious checking from the registry keys. Automatic deletion is a key element when we compare registry cleaners.<br><br>Next, there is an simple technique to deal with this issue. You are able to install a registry cleaner that you can get it online. This software can help you find out these mistakes inside a computer plus clean them. It furthermore could figure out these malware plus other threats that influence the speed of your computer. So this software will accelerate PC simpler. You can choose 1 of these techniques to speed up you computer. |
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| == Formulation ==
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| The most general formulation of the theorem needs some preparation. Given a [[vector space]] ''V'' over the [[field (mathematics)|field]] '''R''' of [[real number]]s, a [[function (mathematics)|function]] {{nowrap|''f'': ''V'' → '''R'''}} is called [[sublinear function|sublinear]] if
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| :''f''(γ ''x'') = γ ''f''(''x'') for any γ ∈ '''R'''<sub>+</sub> and any ''x'' ∈ ''V'' ([[positive homogeneity]]),
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| :''f''(''x'' + ''y'') ≤ ''f''(''x'') + ''f''(''y'') for any ''x'', ''y'' ∈ ''V'' ([[subadditive function|subadditivity]]).
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| Every [[seminorm]] on ''V'' (in particular, every [[norm (mathematics)|norm]] on ''V'') is sublinear. Other sublinear functions can be useful as well, especially [[Minkowski functional]]s of convex sets.
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| <blockquote style="color:#111111; background:#FFFFFF; padding:1em; border:1px solid #999999">
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| '''Hahn–Banach theorem {{harv|Rudin|1991|loc=Th. 3.2}}.''' If ''p'' : ''V'' → '''R''' is a sublinear function, and φ : ''U'' → '''R''' is a [[linear functional]] on a [[linear subspace]] ''U'' ⊆ ''V'' which is [[dominate (mathematics)|dominated]] by ''p'' on ''U'', i.e.
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| :<math>\phi(x) \leq p(x)\qquad\forall x \in U</math>
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| then there exists a linear extension ψ: ''V'' → '''R''' of φ to the whole space ''V'', ''i.e.'', there exists a linear functional ψ such that
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| :<math>\psi(x)=\phi(x)\qquad\forall x\in U,</math>
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| :<math>\psi(x) \le p(x)\qquad\forall x\in V.</math>
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| </blockquote>
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| <blockquote style="color:#111111; background:#FFFFFF; padding:1em; border:1px solid #999999">
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| '''Alternate Version of Hahn–Banach theorem.''' Set '''K''' = '''R''', '''C''' and let ''V'' be a '''K'''-vector space with a seminorm ''p'' : ''V'' → '''R'''. If φ: ''U'' → '''K''' is a '''K'''-linear functional on a '''K'''-linear subspace ''U'' of ''V'' which is dominated by ''p'' on ''U'' in absolute value,
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| :<math>|\phi(x)|\leq p(x)\qquad\forall x \in U</math>
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| then there exists a linear extension ψ : ''V'' → '''K''' of φ to the whole space ''V'', ''i.e.'', there exists a '''K'''-linear functional ψ such that
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| :<math>\psi(x)=\phi(x)\qquad\forall x\in U,</math>
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| :<math>|\psi(x)| \le p(x)\qquad\forall x\in V.</math>
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| </blockquote>
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| In the complex case of the alternate version, the '''C'''-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector ''x'' ∈ ''U'', we have i''x'' ∈ ''U'' and {{nowrap|φ(i''x'') {{=}} iφ(''x'')}}.
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| The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ. The usual proof for the case of an infinite dimensional space ''V'', uses [[Zorn's lemma]] or, equivalently, the [[axiom of choice]]. It is now known (see section 4.0) that the [[ultrafilter lemma]], which is slightly weaker than the axiom of choice, is actually strong enough.
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| It is possible to relax slightly the sublinearity condition on ''p'', requiring only that (Reed and Simon, 1980):
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| :<math>p(ax+by)\leq|a| \, p(x) + |b| \, p(y),\qquad x,y\in V,\quad |a|+|b|\leq1.</math>
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| This reveals the intimate connection between the Hahn–Banach theorem and [[Convex function|convexity]].
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| The [[Mizar system|Mizar project]] has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the [http://mizar.uwb.edu.pl/JFM/Vol5/hahnban.html HAHNBAN file].
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| == Important consequences ==
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| The theorem has several important consequences, some of which are also sometimes called "Hahn–Banach theorem":
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| * If ''V'' is a normed vector space with linear subspace ''U'' (not necessarily closed) and if {{nowrap|φ : ''U'' → '''K'''}} is continuous and linear, then there exists an extension {{nowrap|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). In other words, in the category of normed vector spaces, the space '''K''' is an [[injective object]].
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| * If ''V'' is a normed vector space with linear subspace ''U'' (not necessarily closed) and if ''z'' is an element of ''V'' not in the [[closure (topology)|closure]] of ''U'', then there exists a continuous linear map {{nowrap|ψ : ''V'' → '''K'''}} with ψ(''x'') = 0 for all ''x'' in ''U'', ψ(''z'') = 1, and {{nowrap begin}}||ψ|| = dist(''z'', ''U'')<sup>−1</sup>{{nowrap end}}.
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| * In particular, if ''V'' is a normed vector space and if ''z'' is any element of ''V'', then there exists a continuous linear map {{nowrap|ψ : ''V'' → '''K'''}} with {{nowrap begin}}ψ(''z'') = ||z||{{nowrap end}} and {{nowrap begin}}||ψ|| ≤ 1{{nowrap end}}. This implies that the [[Reflexive space#Normed spaces|natural injection]] ''J'' from a normed space ''V'' into its [[Reflexive space#Normed spaces|double dual]] ''V′′'' is isometric.
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| == Hahn–Banach separation theorem ==
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| Another version of Hahn–Banach theorem is known as the '''Hahn–Banach separation theorem'''.<ref>Gabriel Nagy, [http://www.math.ksu.edu/~nagy/real-an/ap-e-h-b.pdf Real Analysis] [http://www.math.ksu.edu/~nagy/real-an/ lecture notes]</ref> It has numerous uses in [[convex geometry]],<ref>{{cite journal |first=R. |last=Harvey |first2=H. B. |last2=Lawson |title=An intrinsic characterisation of Kähler manifolds |journal=[[Inventiones Mathematicae|Invent. Math]] |volume=74 |year=1983 |issue=2 |pages=169–198 |doi=10.1007/BF01394312 |ref=harv }}</ref> [[optimization (mathematics)|optimization theory]], and [[mathematical economics#Functional analyis|economics]]. The separation theorem is derived from the original form of the theorem.
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| <blockquote style="color:#111111; background:#FFFFFF; padding:1em; border:1px solid #999999">
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| '''Theorem.''' Set '''K''' = '''R''' or '''C''' and let ''V'' be a [[topological vector space]] over '''K'''. If ''A'', ''B'' are convex, non-empty disjoint subsets of ''V'', then:
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| *If ''A'' is open, then there exists a continuous linear map λ : ''V'' → '''K''' and ''t'' ∈ '''R''' such that Re(λ(''a'')) < ''t'' ≤ Re(λ(''b'')) for all ''a'' ∈ ''A'', ''b'' ∈ ''B''.
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| *If ''V'' is locally convex, ''A'' is compact, and ''B'' closed, then there exists a continuous linear map λ : ''V'' → '''K''' and ''s'', ''t'' ∈ '''R''' such that Re(λ(''a'')) < ''t'' < ''s'' < Re(λ(''b'')) for all ''a'' ∈ ''A'', ''b'' ∈ ''B''.</blockquote>
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| == Geometric Hahn–Banach theorem ==
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| One form of Hahn-Banach theorem is known as the '''Geometric Hahn-Banach Theorem''', or '''Mazur's Theorem'''.<ref>Luenberger, David G. (1969), Optimization by vector space methods, John Wiley & Sons, Inc., ISBN 0471-18117-X</ref>
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| <blockquote style="color:#111111; background:#FFFFFF; padding:1em; border:1px solid #999999">
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| '''Theorem.''' Let ''K'' be a convex set having a nonempty interior in a real normed linear vector space ''X''. Suppose ''V'' is a linear variety in ''X'' containing no interior points of ''K''. Then there is a closed hyperplane in ''X'' containing ''V'' but containing no interior points of ''K''; i.e., there is an element ''x''<sup>''*''</sup> ∈ ''X''<sup>''*''</sup> and a constant ''c'' such that <''v'',''x''<sup>''*''</sup> > = ''c'' for all ''v'' ∈ ''V'' and <''k'',''x''<sup>''*''</sup> > < ''c'' for all ''k'' ∈ '''int''' ''K''.</blockquote>
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| This can be generalized to an arbitrary topological vector space, which need not be localy convex or even Hausdorff, as:<ref>Treves, p. 184</ref>
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| <blockquote style="color:#111111; background:#FFFFFF; padding:1em; border:1px solid #999999"> | |
| '''Theorem.''' Let ''M'' be a vector subspace of the topological vector space ''X''. Suppose ''K'' is a non-empty convex open subset of ''X'' with <math>K \cap M = \emptyset</math>. Then there is a closed hyperplane ''N'' in ''X'' containing ''M'' with <math>K \cap N = \emptyset</math>.</blockquote>
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| == Relation to the axiom of choice ==
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| As mentioned earlier, the [[axiom of choice]] implies the Hahn–Banach theorem. The converse is not true. One way to see that is by noting that the [[ultrafilter lemma]] (or equivalently, the Boolean prime ideal theorem), 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.
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| The Hahn–Banach theorem is equivalent to the following:<ref>{{cite book|first=Eric|last=Schechter|title=Handbook of Analysis and its Foundations|page=620}}</ref>
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| :(*): On every Boolean algebra ''B'' there exists a "probability charge", that is: a nonconstant finitely additive map from ''B'' into [0, 1].
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| (The Boolean prime ideal theorem is easily seen to be equivalent to the statement that there are always probability charges which take only the values 0 and 1.)
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| In ZF, one can show that the Hahn–Banach theorem is enough to derive the existence of a non-Lebesgue measurable set.<ref>{{cite journal |first=M. |last=Foreman |first2=F. |last2=Wehrung |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13812.pdf |title=The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set |journal=Fundamenta Mathematicae |volume=138 |issue= |year=1991 |pages=13–19 |doi= |ref=harv }}</ref> Moreover, the Hahn–Banach theorem implies the [[Banach-Tarski paradox]].<ref>{{cite journal|first=Janusz|last=Pawlikowski|title=The Hahn-Banach theorem implies the Banach-Tarski paradox|journal=Fundamenta Mathematicae|volume=138 |issue= |year=1991 |pages=21–22}}</ref>
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| For [[Separable space|separable]] [[Banach space]]s, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL<sub>0</sub>, a weak subsystem of [[second-order arithmetic]] that takes [[König's Lemma]] as an axiom. In fact, they prove that under a weak set of assumptions, the two are equivalent, an example of [[Reverse mathematics]].<ref>{{cite journal |first=D. K. |last=Brown |first2=S. G. |last2=Simpson |title=Which set existence axioms are needed to prove the separable Hahn–Banach theorem? |journal=Annals of Pure and Applied Logic |volume=31 |issue= |year=1986 |pages=123–144 |doi=10.1016/0168-0072(86)90066-7 |ref=harv }} [http://www.math.psu.edu/simpson/papers/hilbert/node7.html#3 Source of citation].</ref><ref> Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, ISBN 978-0-521-88439-6, MR2517689 </ref>
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| ==Consequences==
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| ===Topological vector spaces===
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| If ''X'' is a topological vector space, not necessarily Hausdorff or locally convex, then there exists a non-zero continuous linear form if and only if ''X'' contains a non-empty, convex, open set ''U'' such that ''U ≠ X''.<ref>Schaefer 1999, p. 47</ref> So if the continuous dual space of ''X'', <math>X^*</math>, is non-trivial then by considering ''X'' with the weak topology induced by <math>X^*</math>, ''X'' becomes a locally convex topological vector space with a non-trivial topology that is weaker than ''X'''s original topology. If in addition, <math>X^*</math> separates points on ''X'' (which means that for each ''x ∈ X'' there is a linear functional in <math>X^*</math> that's non-zero on ''x'') then ''X'' with this weak topology becomes Hausdorff. This soetimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.
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| ===The dual space ''C''[''a'', ''b'']*===
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| We have the following consequence of the Hahn–Banach theorem.
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| <blockquote>'''Proposition'''. Let −∞ < ''a'' < ''b'' < ∞. Then, ''F'' ∈ ''C''[''a'', ''b'']* if and only if there exists a function ρ : [''a'', ''b''] → '''R''' of bounded variation such that
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| :<math>F(u)=\int^b_a u(x)d\rho(x),</math>
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| for all ''u'' ∈ ''C''[''a'', ''b'']. In addition, |''F''| = ''V''(ρ), where ''V''(ρ) denotes the total variation of ρ.</blockquote>
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| == See also ==
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| * [[M. Riesz extension theorem]]
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| * [[Separating axis theorem]]
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| ==Notes==
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| <references />
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| ==References==
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| * {{springer|title=Hahn-Banach theorem|id=p/h046130}}
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| * Lawrence Narici and Edward Beckenstein, "[http://at.yorku.ca/p/a/a/a/16.htm The Hahn–Banach Theorem: The Life and Times]", ''Topology and its Applications'', Volume '''77''' (1997), 193–211.
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| * Lothar M Schmitt, An Equivariant Version of the Hahn-Banach Theorem, [http://www.math.uh.edu/~hjm/vol18-3.html Houston J. of Math. 18 (1992), 429-447]
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| * Michael Reed and Barry Simon, ''Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis,'' Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.
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| * {{Cite isbn|9780070542365}} <!-- Rudin, Walter's Functional Analysis -->
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| * Terence Tao, [http://terrytao.wordpress.com/2007/11/30/the-hahn-banach-theorem-mengers-theorem-and-hellys-theorem The Hahn–Banach theorem, Menger’s theorem, and Helly’s theorem]
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| * {{Cite isbn|9780486453521| pages=136-149, 195-201, 240-252, 335-390, 420-433 }} <!-- Trèves, François's Topological Vector Spaces, Distributions and Kernels -->
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| * Gerd Wittstock, Ein operatorwertiger Hahn-Banach Satz, [http://www.sciencedirect.com/science/article/pii/0022123681900641 J. of Functional Analysis 40 (1981), 127–150]
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| * Eberhard Zeidler, ''Applied Functional Analysis: main principles and their applications'', Springer, 1995.
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Hahn-Banach Theorem}}
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| [[Category:Functional analysis]]
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| [[Category:Theorems in functional analysis]]
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If you have the desire to procedure settings immediately, loading files fast, but the body is logy plus torpid, what would you do? If you are a giant "switchboard" that is deficiency of efficient management system and effective housekeeper, what would we do? If you have send the exact commands to a notice, yet the body cannot perform correctly, what would we do? Yes! You want a full-featured repair registry!
If it happens to be not because big of the issue as we think it happens to be, it will probably be solved conveniently by running a Startup Repair or by System Restore Utility. Again it may be because effortless because running an anti-virus check or cleaning the registry.
It doesn't matter whether you're not especially obvious about what rundll32.exe is. But remember which it plays an significant part inside maintaining the stability of the computers plus the integrity of the program. When several software or hardware could not answer normally to a program operation, comes the rundll32 exe error, which will be caused by corrupted files or missing information inside registry. Usually, error content may shows up at booting or the beginning of running a system.
If you feel we don't have enough money at the time to upgrade, then the greatest choice is to free up some room by deleting a few of the unwelcome files and folders.
There are numerous registry reviver s found on the market now. How do you understand that you to choose? Well, when you bought a car you did several research on it, didn't you? You didn't only go out plus purchase the initial red convertible you saw. The same thing works with registry cleaners. On any search engine, type in "registry cleaner reviews" and they may receive posted for you to read about.
The initial thing you need to do is to reinstall any system which shows the error. It's typical for various computers to have particular programs which require this DLL to show the error when we try plus load it up. If you see a particular program show the error, you need to first uninstall that system, restart a PC plus then resinstall the system again. This must replace the damaged ac1st16.dll file and remedy the error.
We need an option to automatically delete unwanted registry keys. This will protect you hours of laborious checking from the registry keys. Automatic deletion is a key element when we compare registry cleaners.
Next, there is an simple technique to deal with this issue. You are able to install a registry cleaner that you can get it online. This software can help you find out these mistakes inside a computer plus clean them. It furthermore could figure out these malware plus other threats that influence the speed of your computer. So this software will accelerate PC simpler. You can choose 1 of these techniques to speed up you computer.