In mathematics, an LB-space, also written (LB)-space, is a topological vector space X {\displaystyle X} that is a locally convex inductive limit of a countable inductive system ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} of Banach spaces. This means that X {\displaystyle X} is a direct limit of a direct system ( X n , i n m ) {\displaystyle \left(X_{n},i_{nm}\right)} in the category of locally convex topological vector spaces and each X n {\displaystyle X_{n}} is a Banach space. If each of the bonding maps i n m {\displaystyle i_{nm}} is an embedding of TVSs then the LB-space is called a strict LB-space. This means that the topology induced on X n {\displaystyle X_{n}} by X n + 1 {\displaystyle X_{n+1}} is identical to the original topology on X n . {\displaystyle X_{n}.} Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space."
Definition The topology on X {\displaystyle X} can be described by specifying that an absolutely convex subset U {\displaystyle U} is a neighborhood of 0 {\displaystyle 0} if and only if U ∩ X n {\displaystyle U\cap X_{n}} is an absolutely convex neighborhood of 0 {\displaystyle 0} in X n {\displaystyle X_{n}} for every n . {\displaystyle n.}
Properties A strict LB-space is complete, barrelled, and bornological (and thus ultrabornological).
Examples If D {\displaystyle D} is a locally compact topological space that is countable at infinity (that is, it is equal to a countable union of compact subspaces) then the space C c ( D ) {\displaystyle C_{c}(D)} of all continuous, complex-valued functions on D {\displaystyle D} with compact support is a strict LB-space. For any compact subset K ⊆ D , {\displaystyle K\subseteq D,} let C c ( K ) {\displaystyle C_{c}(K)} denote the Banach space of complex-valued functions that are supported by K {\displaystyle K} with the uniform norm and order the family of compact subsets of D {\displaystyle D} by inclusion.
Final topology on the direct limit of finite-dimensional Euclidean spaces Let
R ∞ := { ( x 1 , x 2 , … ) ∈ R N : all but finitely many x i are equal to 0 } , {\displaystyle {\begin{alignedat}{4}\mathbb {R} ^{\infty }~&:=~\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to 0 }}\right\},\end{alignedat}}}
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