View source for Pap adjoint theorem

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p1200401.png" /> is a [[Topological vector space|topological vector space]], a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p1200402.png" /> in X is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p1200405.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p1200406.png" /> sequence if every subsequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p1200407.png" /> has a further subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p1200408.png" /> such that the subseries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p1200409.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004010.png" />-convergent to an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004011.png" />.

A topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004012.png" /> is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004014.png" />-space if every sequence which converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004015.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004016.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004017.png" /> sequence. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004019.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004022.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004023.png" /> bounded if for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004024.png" /> and every scalar sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004025.png" /> converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004026.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004027.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004028.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004029.png" /> sequence.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004031.png" /> be Hausdorff locally convex topological vector spaces (cf. also [[Locally convex space|Locally convex space]]; [[Hausdorff space|Hausdorff space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004032.png" /> be a linear mapping. The domain of the adjoint operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004033.png" />, is defined to be

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004034.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004035.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004036.png" />.

The first adjoint theorem was proved by E. Pap [[#References|[a3]]] for operators defined on pre-Hilbert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004037.png" />-spaces. There exists a pre-Hilbert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004038.png" />-space which is not a Hilbert space, [[#References|[a2]]]. A generalization of the adjoint theorem for normed spaces was given in [[#References|[a1]]], [[#References|[a4]]]. It reads as follows.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004039.png" /> be a normed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004040.png" />-space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004041.png" /> be a [[Normed space|normed space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004042.png" /> be a [[Linear operator|linear operator]]. Then the adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004043.png" /> is a bounded linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004044.png" />.

In the proofs of all these theorems, so-called diagonal theorems were used (cf. also [[Diagonal theorem|Diagonal theorem]]). As a simple consequence, a proof of the closed-graph theorem without the Baire category argumentation was obtained, [[#References|[a3]]], [[#References|[a4]]], [[#References|[a6]]], [[#References|[a8]]], [[#References|[a9]]].

There is a locally convex generalization of the adjoint theorem [[#References|[a5]]], [[#References|[a7]]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004045.png" /> is sequentially continuous with respect to the relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004046.png" />- (weak) topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004047.png" /> and the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004048.png" /> of uniform convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004049.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004050.png" />-convergent sequences. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004051.png" /> is bounded with respect to these topologies.

A special case is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004052.png" /> is a normed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004053.png" />-space. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004054.png" /> maps weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004055.png" /> bounded subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004056.png" /> to norm-bounded subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004057.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120040/p12004058.png" /> is norm-bounded.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Antosik,   Swartz, C.,   "Matrix methods in analysis" , ''Lecture Notes Math.'' , '''1113''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Kliś,   "An example of non-complete normed (K)-space"  ''Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys.'' , '''26'''  (1976)  pp. 415–420</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Pap,   "Functional analysis with K-convergence" , ''Proc. Conf. Convergence, Bechyne, Czech.'' , Akad. Berlin  (1984)  pp. 245–250</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Pap,   "The adjoint operator and K-convergence"  ''Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.'' , '''15''' :  2  (1985)  pp. 51–56</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Pap,   "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Pap,   C. Swartz,   "The closed graph theorem for locally convex spaces"  ''Boll. Un. Mat. Ital.'' , '''7''' :  4-B  (1990)  pp. 109–111</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Pap,   C. Swartz,   "A locally convex version of adjoint theorem"  ''Univ. u Novom Sadu Zb. Rad. Prirod. - Mat. Fak. Ser. Mat.'' , '''24''' :  2  (1994)  pp. 63–68</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C. Swartz,   "The closed graph theorem without category"  ''Bull. Austral. Math. Soc.'' , '''36'''  (1987)  pp. 283–288</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  C. Swartz,   "Introduction to functional analysis" , M. Dekker  (1992)</TD></TR></table>