Russian Mathematics (Iz. VUZ)
Vol. 47, No. 9, pp.43{49, 2003
Izvestiya VUZ. Matematika
UDC 517.98
SPACES WITH GRAPH NORM
AND STRENGTHENED SOBOLEV SPACES. II
M.R. Timerbayev
1. Spaces of vector-functions with values in spaces with graph norm
This paper is a continuation of 1]. Recall the basic notions and notation. We denote by , topological vector spaces (t. v. s.). Unless otherwise specied, for a t. v. s. U , the inclusion U is
always understood not only in the set-theoretical sense but in the topological sense as well. Thus,
for B -spaces, the equality U = V also means that the norms of these spaces are equivalent. Note
that if B -spaces U V are such that V is a subset of U , then V is continuously embedded
into U . This follows from the Banach closed graph theorem applied to the identity mapping.
Denote by L( ) the set of linear continuous mappings from to . Let 2 L( ) and
U . Then, obviously, 2 L(U ) (here and everywhere below the restriction of to U is
denoted by the same symbol ). If U is a B -space, then the linear set (U ) (the image of U
under ) endowed with the quotient norm
kxk (U ) = inf fkukU : u 2 U u = xg
is, obviously, a B -space isometric to the quotient space U=(ker \ U ), and (U ) , 2
L(U (U )). Besides that, if a B -space X is continuously embedded into , then in order that map U continuously to X , it is sucient (and, clearly, necessary) that (U ) be a subset of X . If maps a B -space U continuously onto a B -space X , then the quotient norm of the space (U ) is
equivalent to the norm of X .
The operator 2 L(U X ) is called a retraction if there exists an operator 2 L(X U ), called
a coretraction, such that
x = x 8x 2 X:
Let 2 L( ), and let U , X be B -spaces. Consider the space
(U X ) = fu 2 U : u 2 X g
def
endowed with the graph norm
kuk
UX ) = kukU
+ kukX :
(1)
From the denition of the norm (1) and from the completeness of the spaces U , X it follows that
(U X ) is a B -space continuously embedded into U . Moreover, (U X ) = U if and only if (U ) is
a subset of X .
(
The work was supported by the Russian Foundation for Basic Research (grants 01-01-00616, 03-01-00380)
and by the Ministry of Education of Russian Federation (grant no. E02-1.0-189 for fundamental research in
the eld of natural and exact sciences).
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