View source for Differentiable vector in a representation space

''$V$ of a representation $T$ of a Lie group $G$''

A vector $\xi \in V$ for which the mapping
$$
g \mapsto T(g) \xi
$$
is an infinitely-differentiable (of class $C^\infty$) vector function on $G$ with values in $V$. For the vector function $f : G \rightarrow V$ to be differentiable, a necessary condition (and, in the case of a locally convex quasi-complete space $V$, also a sufficient condition) is that all scalar functions of the type $F \circ f$, where $F$ is a linear continuous functional on $V$, be differentiable [[#References|[1]]]. The Gel'fand–Gårding theorem may be stated as follows: If $T$ is a continuous representation of a [[Lie group]] $G$ in a [[Banach space]] $V$, then the set $V^\infty$ of differentiable vectors is dense in $V$. 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>

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