Topologies on spaces of linear maps

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V ↦ W between two modules (including vector spaces) that preserves the operations of addition and scalar multiplication.

By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like topologies or bornolgies, then one can study the subspace of linear maps that preserve this structure.

Spaces of Continuous Linear Maps

Throughout this section we will assume that $X$ and $Y$ are topological vector spaces and we will let $L(X,Y)$, denote the vector space of all continuous linear maps from $X$ and $Y$.

${\mathcal {G}}$-topologies

We can form topologies on $L(X,Y)$ in the following way. Let ${\mathcal {G}}$ be a set of subsets of $X$ and let ${\mathcal {N}}$ be a basis of neighborhoods of 0 in $Y$. Then we can define a topology on $L(X,Y)$ by defining the neighborhoods of 0 in $L_{\mathcal {G}}(X,Y)$ to be

${\mathcal {U}}(G,V)={f\in L(X,Y):f(G)\subset N}$

as G and N range over all $G\in {\mathcal {G}}$ and $N\in {\mathcal {N}}$. This topology is known as the ${\mathcal {G}}$-topology on $L(X,Y)$ or as the topology of uniform convergence on the sets in ${\mathcal {G}}$ and $L(X,Y)$ with this topology is and denoted by $L_{\mathcal {G}}(X,Y)$.

Note that the ${\mathcal {G}}$-topology on $L(X,Y)$ is compatible with the vector space structor of $L(X,Y)$ (that is, $L_{\mathcal {G}}(X,Y)$ is a topological vector space) if and only if for all $G\in {\mathcal {G}}$ and all $f\in L(X,Y)$ the set $f(G)$ is bounded in $Y$; we will assume this to be the case for the rest of the article. Note that in particular, this is the case if ${\mathcal {G}}$ consists of (von-Neumann) bounded subsets of $X$.

Examples

• The topology of pointwise convergence is the ${\mathcal {G}}$-topology on $L(X,Y)$ induced by ${\mathcal {G}}=\{G\subset X:G{\text{ is finite}}\}$. $L(X,Y)$ with this topology is denoted by $L_{\sigma }(X,Y)$.
• The topology of convex compact convergence or the topology of uniform convergence on compact convex sets is the ${\mathcal {G}}$-topology on $L(X,Y)$ induced by ${\mathcal {G}}=\{G\subset X:G{\text{ is convex and compact}}\}$. $L(X,Y)$ with this topology is denoted by $L_{\gamma }(X,Y)$.
• The topology of compact convergence or the topology of uniform convergence on compact sets is the ${\mathcal {G}}$-topology on $L(X,Y)$ induced by ${\mathcal {G}}=\{G\subset X:G{\text{ is compact}}\}$. $L(X,Y)$ with this topology is denoted by $L_{c}(X,Y)$.
• The topology of bounded convergence or the topology of uniform convergence on bounded sets is the ${\mathcal {G}}$-topology on $L(X,Y)$ induced by ${\mathcal {G}}=\{G\subset X:G{\text{ is bounded}}\}$. $L(X,Y)$ with this topology is denoted by $L_{b}(X,Y)$.

Properties

• If $Y$ is locally convex then so is $L_{\mathcal {G}}(X,Y)$.
• If $Y$ is Hausdorff and $\bigcup _{G\in {\mathcal {G}}}G$ is dense in $X$ then $L_{\mathcal {G}}(X,Y)$ is Hausdorff.

See also

• Bounded linear operator
• Operator norm

References

• Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
• H.H. Schaefer (1970). Topological Vector Spaces. GTM. Vol. 3. Springer-Verlag. pp. 61–63. ISBN 0-387-05380-8.
• Tr\`{e}ves, Fran\c{c}ois (1995). Topological Vector Spaces, Distributions and Kernels. Academic Press, Inc. pp. 136–149, 195–201, 240–252, 335–390, 420–433. ISBN 0-486-45352-9.
• S.M. Khaleelulla (1982). Counterexamples in Topological Vector Spaces. GTM. Vol. 936. Springer-Verlag. pp. 29–33, 49, 104. ISBN 978-3-540-11565-6.