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In
functional analysis , a subset of a
topological vector space (TVS) is called a barrel or a barrelled set if it is closed
convex
balanced and
absorbing .
Barrelled sets play an important role in the definitions of several classes of topological vector spaces, such as
barrelled spaces .
Definitions
Let
X
{\displaystyle X}
be a
topological vector space (TVS).
A subset of
X
{\displaystyle X}
is called a barrel if it is closed
convex
balanced and
absorbing in
X
.
{\displaystyle X.}
A subset of
X
{\displaystyle X}
is called
bornivorous and a bornivore if it
absorbs every
bounded subset of
X
.
{\displaystyle X.}
Every bornivorous subset of
X
{\displaystyle X}
is necessarily an absorbing subset of
X
.
{\displaystyle X.}
Let
B
0
⊆
X
{\displaystyle B_{0}\subseteq X}
be a subset of a topological vector space
X
.
{\displaystyle X.}
If
B
0
{\displaystyle B_{0}}
is a
balanced
absorbing subset of
X
{\displaystyle X}
and if there exists a sequence
(
B
i
)
i
=
1
∞
{\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}
of balanced absorbing subsets of
X
{\displaystyle X}
such that
B
i
+
1
+
B
i
+
1
⊆
B
i
{\displaystyle B_{i+1}+B_{i+1}\subseteq B_{i}}
for all
i
=
0
,
1
,
…
,
{\displaystyle i=0,1,\ldots ,}
then
B
0
{\displaystyle B_{0}}
is called a suprabarrel in
X
,
{\displaystyle X,}
where moreover,
B
0
{\displaystyle B_{0}}
is said to be a(n):
bornivorous suprabarrel if in addition every
B
i
{\displaystyle B_{i}}
is a closed and
bornivorous subset of
X
{\displaystyle X}
for every
i
≥
0.
{\displaystyle i\geq 0.}
ultrabarrel if in addition every
B
i
{\displaystyle B_{i}}
is a
closed subset of
X
{\displaystyle X}
for every
i
≥
0.
{\displaystyle i\geq 0.}
bornivorous ultrabarrel if in addition every
B
i
{\displaystyle B_{i}}
is a closed and bornivorous subset of
X
{\displaystyle X}
for every
i
≥
0.
{\displaystyle i\geq 0.}
In this case,
(
B
i
)
i
=
1
∞
{\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}
is called a defining sequence for
B
0
.
{\displaystyle B_{0}.}
Properties
Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.
Examples
See also
References
Bibliography
Hogbe-Nlend, Henri (1977). Bornologies and functional analysis . Amsterdam: North-Holland Publishing Co. pp. xii+144.
ISBN
0-7204-0712-5 .
MR
0500064 .
Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces .
Lecture Notes in Mathematics . Vol. 936. Berlin, Heidelberg, New York:
Springer-Verlag .
ISBN
978-3-540-11565-6 .
OCLC
8588370 .
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces . Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
ISBN
978-1584888666 .
OCLC
144216834 .
H.H. Schaefer (1970). Topological Vector Spaces .
GTM . Vol. 3.
Springer-Verlag .
ISBN
0-387-05380-8 .
Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces .
GTM . Vol. 936. Berlin Heidelberg:
Springer-Verlag . pp. 29–33, 49, 104.
ISBN
9783540115656 .
Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis . Mathematical Surveys and Monographs.
American Mathematical Society .
ISBN
9780821807804 .
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