In mathematics, an LF-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system
(
V
n
,
i
n
m
)
of Fréchet spaces. This means that V is a direct limit of the system
(
V
n
,
i
n
m
)
in the category of locally convex topological vector spaces and each
V
n
is a Fréchet space.
Some authors restrict the term LF-space to mean that V is a strict locally convex inductive limit, which means that the topology induced on
V
n
by
V
n
+
1
is identical to the original topology on
V
n
.
The topology on V can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if
U
∩
V
n
is an absolutely convex neighborhood of 0 in
V
n
for every n.
An LF-space is barrelled and bornological (and thus ultrabornological).
A typical example of an LF-space is,
C
c
∞
(
R
n
)
, the space of all infinitely differentiable functions on
R
n
with compact support. The LF-space structure is obtained by considering a sequence of compact sets
K
1
⊂
K
2
⊂
…
⊂
K
i
⊂
…
⊂
R
n
with
⋃
i
K
i
=
R
n
and for all i,
K
i
is a subset of the interior of
K
i
+
1
. Such a sequence could be the balls of radius i centered at the origin. The space
C
c
∞
(
K
i
)
of infinitely differentiable functions on
R
n
with compact support contained in
K
i
has a natural Fréchet space structure and
C
c
∞
(
R
n
)
inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets
K
i
.
With this LF-space structure,
C
c
∞
(
R
n
)
is known as the space of test functions, of fundamental importance in the theory of distributions.
