only if its graph is closed in . Every bounded linear operator from an LF-space into another TVS is continuous. If is an LF-space defined by a sequence then the strong dual space of is a Fréchet space if and only if all are normable. Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space. Examples Space of smooth compactly supported functions A typical example of an LF-space is, , the space of all infinitely differentiable functions on with compact support. The LF-space structure is obtained by considering a sequence of compact sets with and for all i, is a subset of the interior of . Such a sequence could be the balls of radius i centered at the origin. The space of infinitely differentiable functions on with compact support contained in has a natural Fréchet space structure and inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets . With this LF-space structure, is known as the space of test functions, of fundamental importance in the theory of distributions. Direct limit of finite-dimensional spaces Suppose that for every positive integer , and for , consider Xm as a vector subspace of via the canonical embedding defined by . Denote the resulting LF-space by . Since any TVS topology on makes continuous the inclusions of the Xm's into , the latter space has the maximum among all TVS topologies on an -vector space with countable Hamel dimension. It is a LC topology, associated with the family of all seminorms on . Also, the TVS inductive limit topology of coincides with the topological inductive limit; that is, the direct limit of the finite dimensional spaces in the category TOP and in the category TVS coincide. The continuous dual space of is equal to the algebraic dual space of , that is the space of all real valued sequences and the weak topology on is equal to the strong topology on (i.e. ).. In fact, it is the unique LC topology on whose topological dual space is X. Furthermore, the canonical map of into the continuous dual space of is surjective. See also DF-space Direct limit Final topology F-space LB-space Citations Bibliography Topological vector spaces