# Topology of uniform convergence

Template:More footnotes 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 bornologies, then one can study the subspace of linear maps that preserve this structure.

## Topologies of uniform convergence

Suppose that T be any set and that ${\displaystyle {\mathcal {G}}}$ be collection of subsets of T. Suppose in addition that Y is a topological vector space (not necessarily Hausdorff or locally convex) and that ${\displaystyle {\mathcal {N}}}$ is a basis of neighborhoods of 0 in Y. Then the set of all functions from T into Y, ${\displaystyle Y^{T}}$, can be given a unique translation-invariant topology by defining a basis of neighborhoods of 0 in ${\displaystyle Y^{T}}$, to be

${\displaystyle {\mathcal {U}}(G,N)=\{f\in Y^{T}:f(G)\subseteq N\}}$

as G and N range over all ${\displaystyle G\in {\mathcal {G}}}$ and ${\displaystyle N\in {\mathcal {N}}}$. This topology does not depend on the basis ${\displaystyle {\mathcal {N}}}$ that was chosen and it is known as the topology of uniform convergence on the sets in ${\displaystyle {\mathcal {G}}}$ or as the ${\displaystyle {\mathcal {G}}}$-topology.[1] In practice, ${\displaystyle {\mathcal {G}}}$ usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, ${\displaystyle {\mathcal {G}}}$ is the collection of compact subsets of T (and T is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of T. A set ${\displaystyle {\mathcal {G}}_{1}}$ of ${\displaystyle {\mathcal {G}}}$ is said to be fundamental with respect to ${\displaystyle {\mathcal {G}}}$ if each ${\displaystyle G\in {\mathcal {G}}}$ is a subset of some element in ${\displaystyle {\mathcal {G}}_{1}}$. In this case, the collection ${\displaystyle {\mathcal {G}}}$ can be replaced by ${\displaystyle {\mathcal {G}}_{1}}$ without changing the topology on ${\displaystyle Y^{T}}$.[1]

However, the ${\displaystyle {\mathcal {G}}}$-topology on ${\displaystyle Y^{T}}$ is not necessarily compatible with the vector space structure of ${\displaystyle Y^{T}}$ or of any of its vector subspaces (that is, it is not necessarily a topological vector space topology on ${\displaystyle Y^{T}}$). Suppose that F is a vector subspace ${\displaystyle Y^{T}}$ so that it inherits the subspace topology from ${\displaystyle Y^{T}}$. Then the ${\displaystyle {\mathcal {G}}}$-topology on F is compatible with the vector space structure of F if and only if for every ${\displaystyle G\in {\mathcal {G}}}$ and every fF, f(G) is bounded in Y.[1]

If Y is locally convex then so is the ${\displaystyle {\mathcal {G}}}$-topology on ${\displaystyle Y^{T}}$ and if ${\displaystyle (p_{\alpha })}$ is a family of continuous seminorms generating this topology on Y then the ${\displaystyle {\mathcal {G}}}$-topology is induced by the following family of seminorms: ${\displaystyle p_{G,\alpha }(f)=\sup _{x\in G}p_{\alpha }(f(x))}$, as G varies over ${\displaystyle {\mathcal {G}}}$ and ${\displaystyle \alpha }$ varies over all indices.[2] If Y is Hausdorff and T is a topological space such that ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$ is dense in T then the ${\displaystyle {\mathcal {G}}}$-topology on subspace of ${\displaystyle Y^{T}}$ consisting of all continuous maps is Hausdorff. If the topological space T is also a topological vector space, then the condition that ${\displaystyle \bigcup _{G\in {\mathcal {G}}}G}$ be dense in T can be replaced by the weaker condition that the linear span of this set be dense in T, in which case we say that this set is total in T.[3]

Let H be a subset of ${\displaystyle Y^{T}}$. Then H is bounded in the ${\displaystyle {\mathcal {G}}}$-topology if and only if for every ${\displaystyle G\in {\mathcal {G}}}$, ${\displaystyle \cup _{u\in H}u(G)}$ is bounded in Y.[2]

## 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. If L(X, Y) if given the ${\displaystyle {\mathcal {G}}}$-topology inherited from ${\displaystyle Y^{X}}$ then this space with this topology is denoted by ${\displaystyle L_{\mathcal {G}}(X,Y)}$. The ${\displaystyle {\mathcal {G}}}$-topology on L(X, Y) is compatible with the vector space structure of L(X, Y) if and only if for all ${\displaystyle G\in {\mathcal {G}}}$ and all fL(X, Y) the set f(G) is bounded in Y, which we will assume to be the case for the rest of the article. Note in particular that this is the case if ${\displaystyle {\mathcal {G}}}$ consists of (von-Neumann) bounded subsets of X.

Often, ${\displaystyle {\mathcal {G}}}$ is required to satisfy the following two axioms:

If ${\displaystyle {\mathcal {G}}}$ is a bornology on X. which is often the case, then these two axioms are satisfied.

### Properties

#### Completeness

For the following theorems, suppose that X is a topological vector space and Y is a locally convex Hausdorff spaces and ${\displaystyle {\mathcal {G}}}$ is a collection of bounded subsets of X that satisfies axioms ${\displaystyle {\mathcal {G}}_{1}}$ and ${\displaystyle {\mathcal {G}}_{2}}$ and forms a covering of X.

#### Boundedness

Let X and Y be topological vector space and H be a subset of L(X, Y). Then the following are equivalent:[2]

Furthermore,

### Examples

#### The topology of pointwise convergence Lσ(X, Y)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all finite subsets of X, L(X, Y) will have the weak topology on L(X, Y) or the topology of pointwise convergence and L(X, Y) with this topology is denoted by ${\displaystyle L_{\sigma }(X,Y)}$

The weak-topology on L(X, Y) has the following properties:

#### Compact-convex convergence Lγ(X, Y)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all compact convex subsets of X, L(X, Y) will have the the topology of compact convex convergence or the topology of uniform convergence on compact convex sets L(X, Y) with this topology is denoted by ${\displaystyle L_{\gamma }(X,Y)}$.

#### Compact convergence Lc(X, Y)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all compact subsets of X, L(X, Y) will have the the topology of compact convergence or the topology of uniform convergence on compact sets and L(X, Y) with this topology is denoted by ${\displaystyle L_{c}(X,Y)}$.

The topology of bounded convergence on L(X, Y) has the following properties:

#### Strong dual topology Lb(X, Y)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all bounded subsets of X, L(X, Y) will have the the topology of bounded convergence on X or the topology of uniform convergence on bounded sets and L(X, Y) with this topology is denoted by ${\displaystyle L_{b}(X,Y)}$.

The topology of bounded convergence on L(X, Y) has the following properties:

## G-topologies on the continuous dual induced by X

The continuous dual space of a topological vector space X over the field ${\displaystyle {\mathcal {F}}}$ (which we will assume to be real or complex numbers) is the vector space ${\displaystyle L(X,{\mathcal {F}})}$ and is denoted by ${\displaystyle X^{*}}$ and sometimes by ${\displaystyle X'}$. Given ${\displaystyle {\mathcal {G}}}$, a set of subsets of X, we can apply all of the preceding to this space by using ${\displaystyle Y={\mathcal {F}}}$ and in this case ${\displaystyle X^{*}}$ with this ${\displaystyle {\mathcal {G}}}$-topology is denoted by ${\displaystyle X_{\mathcal {G}}^{*}}$, so that in particular we have the following basic properties:

### Examples

#### The weak topology σ(X*, A) or the weak* topology

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all finite subsets of X, ${\displaystyle X^{*}}$ will have the weak topology on ${\displaystyle X^{*}}$ more comonly known as the weak* topology or the topology of pointwise convergence, which is denoted by ${\displaystyle \sigma (X^{*},X)}$ and ${\displaystyle X^{*}}$ with this topology is denoted by ${\displaystyle X_{\sigma }^{*}}$ or by ${\displaystyle X_{\sigma (X^{*},X)}^{*}}$ if there may be ambiguity.

The ${\displaystyle \sigma (X^{*},X)}$ topology has the following properties:

#### Compact-convex convergence γ(X*, X)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all compact convex subsets of X, ${\displaystyle X^{*}}$ will have the the topology of compact convex convergence or the topology of uniform convergence on compact convex sets, which is denoted by ${\displaystyle \gamma (X^{*},X)}$ and ${\displaystyle X^{*}}$ with this topology is denoted by ${\displaystyle X_{\gamma }^{*}}$ or by ${\displaystyle X_{\gamma (X^{*},X)}^{*}}$.

#### Compact convergence c(X*, X)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all compact subsets of X, ${\displaystyle X^{*}}$ will have the the topology of compact convergence or the topology of uniform convergence on compact sets, which is denoted by ${\displaystyle c(X^{*},X)}$ and ${\displaystyle X^{*}}$ with this topology is denoted by ${\displaystyle X_{c}^{*}}$ or by ${\displaystyle X_{c(X^{*},X)}^{*}}$.

#### Precompact convergence

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all precompact subsets of X, ${\displaystyle X^{*}}$ will have the the topology of precompact convergence or the topology of uniform convergence on precompact sets.

#### Mackey topology τ(X*, X)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all convex balanced weakly compact subsets of X, ${\displaystyle X^{*}}$ will have the Mackey topology on ${\displaystyle X^{*}}$ or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by ${\displaystyle \tau (X^{*},X)}$ and ${\displaystyle X^{*}}$ with this topology is denoted by ${\displaystyle X_{\tau (X^{*},X)}^{*}}$.

#### Strong dual topology b(X*, X)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all bounded subsets of X, ${\displaystyle X^{*}}$ will have the the topology of bounded convergence on X or the topology of uniform convergence on bounded sets or the strong dual topology on ${\displaystyle X^{*}}$, which is denoted by ${\displaystyle b(X^{*},X)}$ and ${\displaystyle X^{*}}$ with this topology is denoted by ${\displaystyle X_{b}^{*}}$ or by ${\displaystyle X_{b(X^{*},X)}^{*}}$. Due to its importance, the continuous dual space of ${\displaystyle X_{b}^{*}}$, which is commonly denoted by ${\displaystyle X^{**}}$ so that ${\displaystyle (X_{b}^{*})^{*}=X^{**}}$.

The ${\displaystyle b(X^{*},X)}$ topology has the following properties:

#### Mackey topology τ(X*, X**)

By letting ${\displaystyle {\mathcal {G''}}}$ be the set of all convex balanced weakly compact subsets of ${\displaystyle X^{**}=(X_{b}^{*})^{*}}$, ${\displaystyle X^{*}}$ will have the Mackey topology on ${\displaystyle X^{*}}$ induced by ${\displaystyle X^{**}}$' or the topology of uniform convergence on convex balanced weakly compact subsets of ${\displaystyle X^{**}}$, which is denoted by ${\displaystyle \tau (X^{*},X^{**})}$ and ${\displaystyle X^{*}}$ with this topology is denoted by ${\displaystyle X_{\tau (X^{*},X^{**})}^{*}}$.

#### Other examples

• The topology of uniform convergence on convex balanced complete bounded subsets of X.
• The topology of uniform convergence on convex balanced infracomplete bounded subsets of X.

## G-topologies on X induced by the continuous dual

There is a canonical map from X into ${\displaystyle (X_{\sigma }^{*})^{*}}$ which maps an element ${\displaystyle x\in X}$ to the following map: ${\displaystyle x'\in X^{*}\mapsto \langle x',x\rangle }$. By using this canonical map we can identify X as being contained in the continuous dual of ${\displaystyle X_{\sigma }^{*}}$ (that is, continued in ${\displaystyle (X_{\sigma }^{*})^{*}}$). In fact, this canonical map is onto, which means that ${\displaystyle X=(X_{\sigma }^{*})^{*}}$ so that we can through this canonical isomorphism think of X as the continuous dual space of ${\displaystyle X_{\sigma }^{*}}$. Note that it is a common convention that if an equal sign appears between two sets which are clearly not equal, then the equality really means that the sets are isomorphic through some canonical map.

Since we are now regarding X as the continuous dual space of ${\displaystyle X_{\sigma }^{*}}$, we can look at sets of subsets of ${\displaystyle X_{\sigma }^{*}}$, say ${\displaystyle {\mathcal {G'}}}$ and construct a dual space topology on the dual of ${\displaystyle X_{\sigma }^{*}}$, which is X. * A basis of neighborhoods of 0 for ${\displaystyle X_{\mathcal {G'}}}$ is formed by the Polar sets ${\displaystyle G'^{\circ }:=\{x\in X:\sup _{x'\in G'}|\langle x',x\rangle |\leq 1\}}$ as ${\displaystyle G'}$ varies over ${\displaystyle {\mathcal {G'}}}$.

### Examples

#### The weak topology σ(X, X*)

By letting ${\displaystyle {\mathcal {G'}}}$ be the set of all finite subsets of ${\displaystyle X'}$, X will have the weak topology or the topology of pointwise convergence on ${\displaystyle X^{*}}$, which is denoted by ${\displaystyle \sigma (X,X^{*})}$ and X with this topology is denoted by ${\displaystyle X_{\sigma }}$ or by ${\displaystyle X_{\sigma (X,X^{*})}}$ if there may be ambiguity.

#### Convergence on equicontinuous sets ε(X, X*)

By letting ${\displaystyle {\mathcal {G'}}}$ be the set of all equicontinuous subsets ${\displaystyle X^{*}}$, X will have the the topology of uniform convergence on equicontinuous subsets of ${\displaystyle X^{*}}$, which is denoted by ${\displaystyle \epsilon (X,X^{*})}$ and X with this topology is denoted by ${\displaystyle X_{\epsilon }}$ or by ${\displaystyle X_{\epsilon (X,X^{*})}}$.

#### Mackey topology τ(X, X*)

By letting ${\displaystyle {\mathcal {G'}}}$ be the set of all convex balanced weakly compact subsets of ${\displaystyle X^{*}}$, X will have the Mackey topology on X or the topology of uniform convergence on convex balanced weakly compact subsets of ${\displaystyle X^{*}}$, which is denoted by ${\displaystyle \tau (X,X^{*})}$ and X with this topology is denoted by ${\displaystyle X_{\tau }}$ or by ${\displaystyle X_{\tau (X,X^{*})}}$.

#### Bounded convergence b(X, X*)

By letting ${\displaystyle {\mathcal {G}}}$ be the set of all bounded subsets of X, ${\displaystyle X^{*}}$ will have the the topology of bounded convergence or the topology of uniform convergence on bounded sets, which is denoted by ${\displaystyle b(X,X^{*})}$ and ${\displaystyle X^{*}}$ with this topology is denoted by ${\displaystyle X_{b}^{*}}$ or by ${\displaystyle X_{b(X,X^{*})}^{*}}$.

### The Mackey–Arens theorem

Let X be a vector space and let Y be a vector subspace of the algebraic dual of X that separates points on X. Any locally convex Hausdorff topological vector space (TVS) topology on X with the property that when X is equipped with this topology has Y as its the continuous dual space is said to be compatible with duality between X and Y. If we give X the weak topology ${\displaystyle \sigma (X,Y)}$ then ${\displaystyle X_{\sigma (X,Y)}}$ is a Hausdorff locally convex topological vector space (TVS) and ${\displaystyle \sigma (X,Y)}$ is compatible with duality between X and Y (i.e. ${\displaystyle X_{\sigma (X,Y)}^{*}=(X_{\sigma (X,Y)})^{*}=Y}$). We can now ask the question: what are all of the locally convex Hausdorff TVS topologies that we can place on X that are compatible with duality between X and Y? The answer to this question is called the Mackey–Arens theorem:[9]

Theorem. Let X be a vector space and let ${\displaystyle {\mathcal {T}}}$ be a locally convex Hausdorff topological vector space topology on X. Let ${\displaystyle X^{*}}$ denote the continuous dual space of X and let ${\displaystyle X_{\mathcal {T}}}$ denote X with the topology ${\displaystyle {\mathcal {T}}}$. Then the following are equivalent: Template:Ordered list And furthermore, Template:Ordered list

## G-H-topologies on spaces of bilinear maps

We will let ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ denote the space of separately continuous bilinear maps and ${\displaystyle B(X,Y;Z)}$ denote its subspace the space of continuous bilinear maps, where ${\displaystyle X,Y}$ and ${\displaystyle Z}$ are topological vector space over the same field (either the real or complex numbers). In an analogous way to how we placed a topology on L(X, Y) we can place a topology on ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ and ${\displaystyle B(X,Y;Z)}$.

Let ${\displaystyle {\mathcal {G}}}$ be a set of subsets of X, ${\displaystyle {\mathcal {H}}}$ be a set of subsets of Y. Let $\displaystyle \mathcal{G} × \mathcal{H}$ denote the collection of all sets G × H where ${\displaystyle G\in {\mathcal {G}}}$, ${\displaystyle H\in {\mathcal {H}}}$. We can place on $\displaystyle Z^{X × Y}$ the $\displaystyle \mathcal{G} × \mathcal{H}$ -topology, and consequently on any of its subsets, in particular on ${\displaystyle B(X,Y;Z)}$ and on ${\displaystyle {\mathcal {B}}(X,Y;Z)}$. This topology is known as the ${\displaystyle {\mathcal {G}}-{\mathcal {H}}}$-topology or as the topology of uniform convergence on the products ${\displaystyle G\times H}$ of ${\displaystyle {\mathcal {G}}\times {\mathcal {H}}}$.

However, as before, this topology is not necessarily compatible with the vector space structure of ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ or of ${\displaystyle B(X,Y;Z)}$ without the additional requirement that for all bilinear maps, ${\displaystyle b}$ in this space (that is, in ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ or in ${\displaystyle B(X,Y;Z)}$) and for all ${\displaystyle G\in {\mathcal {G}}}$ and ${\displaystyle H\in {\mathcal {H}}}$ the set ${\displaystyle b(G,H)}$ is bounded in X. If both ${\displaystyle {\mathcal {G}}}$ and ${\displaystyle {\mathcal {H}}}$ consist of bounded sets then this requirement is automatically satisfied if we are topologizing ${\displaystyle B(X,Y;Z)}$ but this may not be the case if we are trying to topologize ${\displaystyle {\mathcal {B}}(X,Y;Z)}$. The ${\displaystyle {\mathcal {G}}}$-${\displaystyle {\mathcal {H}}}$-topology on ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ will be compatible with the vector space structure of ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ if both ${\displaystyle {\mathcal {G}}}$ and ${\displaystyle {\mathcal {H}}}$ consists of bounded sets and any of the following conditions hold:

### The ε-topology

Suppose that ${\displaystyle X,Y}$, and ${\displaystyle Z}$ are locally convex spaces and let ${\displaystyle {\mathcal {G}}}$' and ${\displaystyle {\mathcal {H}}}$' be the collections of equicontinuous subsets of ${\displaystyle X^{*}}$ and ${\displaystyle Y^{*}}$, respectively. Then the ${\displaystyle {\mathcal {G}}}$'-${\displaystyle {\mathcal {H}}}$'-topology on ${\displaystyle {\mathcal {B}}(X_{b(X^{*},X)}^{*},Y_{b(X^{*},X)}^{*};Z)}$ will be a topological vector space topology. This topology is called the ε-topology and ${\displaystyle {\mathcal {B}}(X_{b(X^{*},X)}^{*},Y_{b(X^{*},X)};Z)}$ with this topology it is denoted by ${\displaystyle {\mathcal {B}}_{\epsilon }(X_{b(X^{*},X)}^{*},Y_{b(X^{*},X)}^{*};Z)}$ or simply by ${\displaystyle {\mathcal {B}}_{\epsilon }(X_{b}^{*},Y_{b}^{*};Z)}$.

Part of the importance of this vector space and this topology is that it contains many subspace, such as ${\displaystyle {\mathcal {B}}(X_{\sigma (X^{*},X)}^{*},Y_{\sigma (X^{*},X)}^{*};Z)}$, which we denote by ${\displaystyle {\mathcal {B}}(X_{\sigma }^{*},Y_{\sigma }^{*};Z)}$. When this subspace is given the subspace topology of ${\displaystyle {\mathcal {B}}_{\epsilon }}$${\displaystyle (X_{b}^{*},Y_{b}^{*};Z)}$ it is denoted by ${\displaystyle {\mathcal {B}}_{\epsilon }(X_{\sigma }^{*},Y_{\sigma }^{*};Z)}$.

In the instance where Z is the field of these vector spaces ${\displaystyle {\mathcal {B}}(X_{\sigma }^{*},Y_{\sigma }^{*})}$ is a tensor product of X and Y. In fact, if X and Y are locally convex Hausdorff spaces then ${\displaystyle {\mathcal {B}}(X_{\sigma }^{*},Y_{\sigma }^{*})}$ is vector space isomorphic to ${\displaystyle L(X_{\sigma (X^{*},X)}^{*},Y_{\sigma (Y^{*},Y)})}$, which is in tern equal to ${\displaystyle L(X_{\tau (X^{*},X)}^{*},Y)}$.

These spaces have the following properties:

## References

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1. Schaefer (1970) p. 79
2. Schaefer (1970) p. 81
3. Schaefer (1970) p. 80
4. Schaefer (1970) p. 82
5. Schaefer (1970) p. 83
6. Treves pp. 199–200
7. Treves, p. 198
8. Treves, p. 201
9. Treves, pp. 196, 368 - 370