View source for Differentiable vector in a representation space

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318001.png" /> of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318002.png" /> of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318003.png" />''

A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318004.png" /> for which the mapping

<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/d/d031/d031800/d0318005.png" /></td> </tr></table>

is an infinitely-differentiable (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318006.png" />) vector function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318007.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318008.png" />. For the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d0318009.png" /> to be differentiable, a necessary condition (and, in the case of a locally convex quasi-complete space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180010.png" />, also a sufficient condition) is that all scalar functions of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180012.png" /> is a linear continuous functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180013.png" />, be differentiable [[#References|[1]]]. The Gel'fand–Gårding theorem may be stated as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180014.png" /> is a continuous representation of a [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180015.png" /> in a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180016.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180017.png" /> of differentiable vectors is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031800/d03180018.png" />. This theorem has been proved for one-parameter groups in [[#References|[2]]], and for the general case in [[#References|[3]]]. For a generalization of this result to a wide class of representations on locally convex spaces see [[#References|[4]]] and [[#References|[5]]].

The presence of differentiable vectors in the representation space of a Lie group makes it possible to construct a representation of the corresponding Lie algebra, and thus connect the theory of representations of groups with the theory of representations of Lie algebras [[#References|[6]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "Espaces vectoriels topologiques" , Univ. Sao Paulo  (1954)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,   "On one-parameter groups of operators in a normed space"  ''Dokl. Akad. Nauk. SSSR'' , '''25'''  (1939)  pp. 713–718  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Gårding,   "Note on continuous representations of Lie groups"  ''Proc. Nat. Acad. Sci. USA'' , '''33'''  (1947)  pp. 331–332</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.T. Moore,   "Measurable, continuous and smooth vectors for semigroups and group representations" , Amer. Math. Soc.  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.P. Zhelobenko,   "On infinitely differentiable vectors in representation theory"  ''Vestnik Moskov. Univ. Ser. Mat.'' , '''1'''  (1965)  pp. 3–10  (In Russian)  (English summary)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Warner,   "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)</TD></TR></table>