Topologies on spaces of linear maps

In mathematics, a linear map is a mapping X → Y 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 on arbitrary spaces of maps

Throughout we assume the following:

1. T is any non-empty set and 𝒢 is a non-empty collection of subsets of T directed by subset inclusion (i.e. for any G, H ∈ 𝒢 there exists some K ∈ 𝒢 such that G ∪ H ⊆ K).
2. Y is a topological vector space (not necessarily Hausdorff or locally convex) and 𝒩 is a basis of neighborhoods of 0 in Y.
3. Y^T denotes the set of all Y-valued functions with domain T.
4. F is a vector subspace of Y^T (not necessarily consisting of linear maps).

Definition and notation: For any subsets G of X and N of Y, let

"𝒰(G, N) := { f ∈ F : f (G) ⊆ N }.

Basic neighborhoods at the origin

Henceforth assume that G ∈ 𝒢 and N ∈ 𝒩.

Properties

• 𝒰(G, N) is an absorbing subset of F if and only if for all f ∈ F, N absorbs f (G).[1]
• If N is balanced then so is 𝒰(G, N).[1]
• If N is convex then so is 𝒰(G, N).

Algebraic relations

• For any scalar s, s𝒰(G, N) = 𝒰(G, sN); so in particular, -𝒰(G, N) = 𝒰(G, -N).[1]
• 𝒰(G ∪ H, M ∩ N) ⊆ 𝒰(G, M) ∩ 𝒰(H, N) for any subsets G and H of X and non-empty subsets M and N of Y.[2] Thus:
  • If M ⊆ N then 𝒰(G, M) ⊆ 𝒰(G, N).[1]
  • If G ⊆ H then 𝒰(H, N) ⊆ 𝒰(G, N).
  • For any M, N ∈ 𝒩 and subsets G, H, K of T, if G ∪ H ⊆ K then 𝒰(K, M ∩ N) ⊆ 𝒰(G, M) ∩ 𝒰(H, N).
• 𝒰(∅, N) = F.
• 𝒰(G, N) - 𝒰(G, N) ⊆ 𝒰(G, N - N).[3]
• 𝒰(G, M) + 𝒰(G, N) ⊆ 𝒰(G, M + N).[2]
• For any family 𝒮 of subsets of T, 𝒰( S, N) = 𝒰(S, N).[3]
• For any family ℳ of neighborhoods of 0 in Y, 𝒰(G, M) = 𝒰(G, M).[3]

𝒢-topology

Then the set {𝒰(G, N) : G ∈ 𝒢, N ∈ 𝒩} forms a neighborhood basis[4] at the origin for a unique translation-invariant topology on F, where this topology is not necessarily a vector topology (i.e. it might not make F into a TVS). This topology does not depend on the neighborhood basis 𝒩 that was chosen and it is known as the topology of uniform convergence on the sets in 𝒢 or as the 𝒢-topology.[5] However, this name is frequently changed according to the types of sets that make up 𝒢 (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details[6]).

A subset 𝒢₁ of 𝒢 is said to be fundamental with respect to 𝒢 if each G ∈ 𝒢 is a subset of some element in 𝒢₁. In this case, the collection 𝒢 can be replaced by 𝒢₁ without changing the topology on F.[5] One may also replace 𝒢 with the collection of all subsets of all finite unions of elements of 𝒢 without changing the resulting 𝒢-topology on F.[3]

Definition:[2] Call a subset B of T F-bounded if f (B) is a bounded subset of Y for every f ∈ F.

Theorem[5][2] — The 𝒢-topology on F is compatible with the vector space structure of F if and only if every G ∈ 𝒢 is F-bounded; that is, if and only if for every G ∈ 𝒢 and every f ∈ F, f (G) is bounded in Y.

Nets and uniform convergence

Definition:[2] Let f ∈ F and let f_• = (f_i)_{i ∈ I} be a net in F. Then for any subset G of T, say that f_• converges uniformly to f on G if for every N ∈ 𝒩 there exists some i₀ ∈ I such that for every i ∈ I satisfying i ≥ i₀, f_i - f ∈ 𝒰(G, N) (or equivalently, f_i(g) - f (g) ∈ N for every g ∈ G).

Theorem[2] — If f ∈ F and if f_• = (f_i)_{i ∈ I} is a net in F, then f_• → f in the 𝒢-topology on F if and only if for every G ∈ 𝒢, f_• converges uniformly to f on G.

Inherited properties

Local convexity

If Y is locally convex then so is the 𝒢-topology on F and if (p_i)_{i ∈ I} is a family of continuous seminorms generating this topology on Y then the 𝒢-topology is induced by the following family of seminorms:

p_G,i( f ) = p_i( f(x)),

as G varies over 𝒢 and i varies over I.[7]

Hausdorffness

If Y is Hausdorff and T = G then the 𝒢-topology on F is Hausdorff.[2]

Suppose that T is a topological space. If Y is Hausdorff and F is the vector subspace of Y^T consisting of all continuous maps that are bounded on every G ∈ 𝒢 and if G is dense in T then the 𝒢-topology on F is Hausdorff.

Boundedness

A subset H of F is bounded in the 𝒢-topology if and only if for every G ∈ 𝒢, H(G) := h(G) is bounded in Y.[7]

Examples of 𝒢-topologies

Pointwise convergence

If we let 𝒢 be the set of all finite subsets of T then the 𝒢-topology on F is called the topology of pointwise convergence. The topology of pointwise convergence on F is identical to the subspace topology that F inherits from Y^T when Y^T is endowed with the usual product topology.

If X is a non-trivial completely regular Hausdorff topological space and C(X) is the space of all real (or complex) valued continuous functions on X, the topology of pointwise convergence on C(X) is metrizable if and only if X is countable.[2]

𝒢-topologies on spaces of continuous linear maps

Throughout this section we will assume that X and Y are topological vector spaces. 𝒢 will be a non-empty collection of subsets of X directed by inclusion.

Notation: L(X; Y) will denote the vector space of all continuous linear maps from X into Y. If L(X; Y) is given the 𝒢-topology inherited from Y^X then this space with this topology is denoted by L_𝒢(X, Y).

Notation: The continuous dual space of a topological vector space X over the field 𝔽 (which we will assume to be real or complex numbers) is the vector space L(X; 𝔽) and is denoted by X'.

The 𝒢-topology on L(X; Y) is compatible with the vector space structure of L(X; Y) if and only if for all G ∈ 𝒢 and all f ∈ L(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 𝒢 consists of (von-Neumann) bounded subsets of X.

Assumptions on 𝒢

Assumptions that guarantee a vector topology

Assumption (𝒢 is directed): 𝒢 will be a non-empty collection of subsets of X directed by (subset) inclusion. That is, for any G, H ∈ 𝒢, there exists K ∈ 𝒢 such that G ∪ H ⊆ K.

The above assumption guarantees that the collection of sets 𝒰(G, N) forms a filter base. The next assumption will guarantee that the sets 𝒰(G, N) are balanced. Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdonsome.

Assumption (N ∈ 𝒩 are balanced): 𝒩 is a neighborhoods basis of 0 in Y that consists entirely of balanced sets.

The following assumption is very commonly made because it will guarantee that each set 𝒰(G, N) is absorbing in L(X; Y).

Assumption (G ∈ 𝒢 are bounded): 𝒢 is assumed to consist entirely of bounded subsets of X.

Other possible assumptions

The next theorem gives ways in which 𝒢 can be modified without changing the resulting 𝒢-topology on Y.

Theorem[1] — Let 𝒢 be a non-empty collection of bounded subsets of X. Then the 𝒢-topology on L(X; Y) is not altered if 𝒢 is replaced by any of the following collections of (also bounded) subsets of X:

1. all subsets of all finite unions of sets in 𝒢;
2. all scalar multiples of all sets in 𝒢;
3. all finite Minkowski sums of sets in 𝒢;
4. the balanced hull of every set in 𝒢;
5. the closure of every set in 𝒢;

and if X and Y are locally convex, then we may add to this list:

6. the closed convex balanced hull of every set in 𝒢.

Common assumptions

Some authors (e.g. Narici) require that 𝒢 satisfy the following condition, which implies, in particular, that 𝒢 is directed by subset inclusion:

𝒢 is assumed to be closed with respect to the formation of subsets of finite unions of sets in 𝒢 (i.e. every subset of every finite union of sets in 𝒢 belongs to 𝒢).

Some authors (e.g. Trèves) require that 𝒢 be directed under subset inclusion and that it satisfy the following condition:

If G ∈ 𝒢 and s is a scalar then there exists a H ∈ 𝒢 such that sG ⊆ H.

If 𝒢 is a bornology on X, which is often the case, then these axioms are satisfied. If 𝒢 is a saturated family of bounded subsets of X then these axioms are also satisfied.

Properties

Hausdorffness

Definition:[8] If T is a TVS then we say that 𝒢 is total in T if the linear span of G is dense in T.

If F is the vector subspace of Y^T consisting of all continuous linear maps that are bounded on every G ∈ 𝒢, then the 𝒢-topology on F is Hausdorff if Y is Hausdorff and 𝒢 is total in T.[1]

Completeness

For the following theorems, suppose that X is a topological vector space and Y is a locally convex Hausdorff spaces and 𝒢 is a collection of bounded subsets of X that covers X, is directed by subset inclusion, and satisfies the following condition: if G ∈ 𝒢 and s is a scalar then there exists a H ∈ 𝒢 such that sG ⊆ H.

• L_𝒢(X; Y) is complete if
  1. X is locally convex and Hausdorff,
  2. Y is complete, and
  3. whenever u : X → Y is a linear map then u restricted to every set G ∈ 𝒢 is continuous implies that u is continuous,
• If X is a Mackey space then L_𝒢(X; Y)is complete if and only if both ${\displaystyle X_{\mathcal {G}}^{\prime }}$ and Y are complete.
• If X is barrelled then L_𝒢(X; Y) is Hausdorff and quasi-complete.
• Let X and Y be TVSs with Y quasi-complete and assume that (1) X is barreled, or else (2) X is a Baire space and X and Y are locally convex. If 𝒢 covers X then every closed equicontinuous subset of L(X; Y) is complete in L_𝒢(X; Y) and L_𝒢(X; Y) is quasi-complete.[9]
• Let X be a bornological space, Y a locally convex space, and 𝒢 a family of bounded subsets of X such that the range of every null sequence in X is contained in some G ∈ 𝒢. If Y is quasi-complete (resp. complete) then so is L_𝒢(X; Y).[10]

Boundedness

Let X and Y be topological vector spaces and H be a subset of L(X; Y). Then the following are equivalent:[7]

1. H is bounded in L_𝒢(X; Y);
2. For every G ∈ 𝒢, H(G) := h(G) is bounded in Y;[7]
3. For every neighborhood V of 0 in Y the set h ⁻¹(V) absorbs every G ∈ 𝒢.

Furthermore,

• If X and Y are locally convex Hausdorff space and if H is bounded in L_𝜎(X; Y) (i.e. pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of X.[11]
• If X and Y are locally convex Hausdorff spaces and if X is quasi-complete (i.e. closed and bounded subsets are complete), then the bounded subsets of L(X; Y) are identical for all 𝒢-topologies where 𝒢 is any family of bounded subsets of X covering X.[11]
• If 𝒢 is any collection of bounded subsets of X whose union is total in X then every equicontinuous subset of L(X; Y) is bounded in the 𝒢-topology.[9]

Examples

𝒢 ⊆ 𝒫(X) ("topology of uniform convergence on ...")	Notation	Name ("topology of...")	Alternative name
finite subsets of X	Lσ(X; Y)	pointwise/simple convergence	topology of simple convergence
precompact subsets of X		precompact convergence
compact convex subsets of X	Lγ(X; Y)	compact convex convergence
compact subsets of X	Lc(X; Y)	compact convergence
bounded subsets of X	Lb(X; Y)	bounded convergence	strong topology

The topology of pointwise convergence L_σ(X; Y)

By letting 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 or the topology of simple convergence and L(X; Y) with this topology is denoted by L_𝜎(X; Y). Unfortunately, this topology is also sometimes called the strong operator topology, which may lead to ambiguity;[1] for this reason, this article will avoid referring to this topology by this name.

Definition: A subset of L(X; Y) is called simply bounded or weakly bounded if it is bounded in L_𝜎(X; Y).

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

• If X is separable (i.e. has a countable dense subset) and if Y is a metrizable topological vector space then every equicontinuous subset H of L_𝜎(X; Y) is metrizable; if in addition Y is separable then so is H.[12]
  • So in particular, on every equicontinuous subset of L(X; Y), the topology of pointwise convergence is metrizable.
• Let Y^X denote the space of all functions from X into Y. If L(X; Y) is given the topology of pointwise convergence then space of all linear maps (continuous or not) X into Y is closed in Y^X.
  • In addition, L(X; Y) is dense in the space of all linear maps (continuous or not) X into Y.
• Suppose X and Y are locally convex. Any simply bounded subset of L(X; Y) is bounded when L(X; Y) has the topology of uniform convergence on convex, balanced, bounded, complete subsets of X. If in addition X is quasi-complete then the families of bounded subsets of L(X; Y) are identical for all 𝒢-topologies on L(X; Y) such that 𝒢 is a family of bounded sets covering X.[11]

Equicontinuous subsets

• The weak-closure of an equicontinuous subset of L(X; Y) is equicontinuous.
• If Y is locally convex, then the convex balanced hull of an equicontinuous subset of L(X; Y) is equicontinuous.
• Let X and Y be TVSs and assume that (1) X is barreled, or else (2) X is a Baire space and X and Y are locally convex. Then every simply bounded subset of L(X; Y) is equicontinuous.[9]
• On an equicontinuous subset H of L(X; Y), the following topologies are identical: (1) topology of pointwise convergence on a total subset of X; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.[9]

Compact convergence L_c(X; Y)

By letting 𝒢 be the set of all compact subsets of X, L(X; Y) will have the topology of compact convergence or the topology of uniform convergence on compact sets and L(X; Y) with this topology is denoted by L_c(X; Y).

The topology of compact convergence on L(X; Y) has the following properties:

• If X is a Fréchet space or a LF-space and if Y is a complete locally convex Hausdorff space then L_c(X; Y) is complete.
• On equicontinuous subsets of L(X; Y), the following topologies coincide:
  • The topology of pointwise convergence on a dense subset of X,
  • The topology of pointwise convergence on X,
  • The topology of compact convergence.
  • The topology of precompact convergence.
• If X is a Montel space and Y is a topological vector space, then L_c(X; Y) and L_b(X; Y) have identical topologies.

Topology of bounded convergence L_b(X; Y)

By letting 𝒢 be the set of all bounded subsets of X, L(X; Y) will have 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 L_b(X; Y).[1]

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

• If X is a bornological space and if Y is a complete locally convex Hausdorff space then L_b(X; Y) is complete.
• If X and Y are both normed spaces then the topology on L(X; Y) induced by the usual operator norm is identical to the topology on L_b(X; Y).[1]
  • In particular, if X is a normed space then the usual norm topology on the continuous dual space X ' is identical to the topology of bounded convergence on X '.
• Every equicontinuous subset of L(X; Y) is bounded in L_b(X; Y).

Polar topologies

Throughout, we assume that X is a TVS.

𝒢-topologies versus polar topologies

If X is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets (e.g. if X is a Hausdorff locally convex space), then a 𝒢-topology on X' (as defined in this article) is a polar topology and conversely, every polar topology if a 𝒢-topology. Consequently, in this case the results mentioned in this article can be applied to polar topologies.

However, if X is a TVS whose bounded subsets are not exactly the same as its weakly bounded subsets, then the notion of "bounded in X" is stronger than the notion of "σ(X, X')-bounded in X" (i.e. bounded in X implies σ(X, X')-bounded in X) so that a 𝒢-topology on X' (as defined in this article) is not necessarily a polar topology. One important difference is that polar topologies are always locally convex while 𝒢-topologies need not be.

Polar topologies have stronger results than the more general topologies of uniform convergence described in this article and we refer the read to the main article: polar topology. We list here some of the most common polar topologies.

List of polar topologies

Suppose that X is a TVS whose bounded subsets are the same as its weakly bounded subsets.

Notation: If 𝛥(Y, X) denotes a polar topology on Y then Y endowed with this topology will be denoted by Y_{𝛥(Y, X)} or simply Y_𝛥 (e.g. for σ(Y, X) we'd have 𝛥 = σ so that Y_{σ(Y, X)} and Y_σ all denote Y with endowed with σ(Y, X)).

𝒢 ⊆ 𝒫(X) ("topology of uniform convergence on ...")	Notation	Name ("topology of...")	Alternative name
finite subsets of X	σ(Y, X) s(Y, X)	pointwise/simple convergence	weak/weak* topology
σ(X, Y)-compact disks	τ(Y, X)		Mackey topology
σ(X, Y)-compact convex subsets	γ(Y, X)	compact convex convergence
σ(X, Y)-compact subsets (or balanced σ(X, Y)-compact subsets)	c(Y, X)	compact convergence
σ(X, Y)-bounded subsets	b(Y, X) 𝛽(Y, X)	bounded convergence	strong topology

𝒢-ℋ-topologies on spaces of bilinear maps

We will let ℬ(X, Y; Z) denote the space of separately continuous bilinear maps and B(X, Y; Z) denote the space of continuous bilinear maps, where X, Y, and 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 ℬ(X, Y; Z) and B(X, Y; Z).

Let 𝒢 (resp. ℋ) be a family of subsets of X (resp. Y) containing at least one non-empty set. Let 𝒢 × ℋ denote the collection of all sets G × H where G ∈ 𝒢, H ∈ ℋ. We can place on Z^{X × Y} the 𝒢 × ℋ-topology, and consequently on any of its subsets, in particular on B(X, Y; Z) and on ℬ(X, Y; Z). This topology is known as the 𝒢-ℋ-topology or as the topology of uniform convergence on the products G × H of 𝒢 × ℋ.

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

• X and Y are barrelled spaces and Z is locally convex.
• X is a F-space, Y is metrizable, and Z is Hausdorff, in which case ℬ(X, Y; Z) = B(X, Y; Z).
• X, Y, and Z are the strong duals of reflexive Fréchet spaces.
• X is normed and Y and Z the strong duals of reflexive Fréchet spaces.

The ε-topology

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

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

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

These spaces have the following properties:

• If X and Y are locally convex Hausdorff spaces then ℬ_ε${\displaystyle \left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is complete if and only if both X and Y are complete.
• If X and Y are both normed (or both Banach) then so is ${\displaystyle {\mathcal {B}}_{\epsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$

See also

• Bornological space – A topological vector space where any bounded linear operator into another space is always continuous
• Bounded linear operator
• Dual system
• Dual topology
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Operator norm – measure of "size" of linear operators
• Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
• Strong dual space – Continuous dual space endowed with the topology of uniform convergence on bounded sets
• Strong topology (polar topology) – Dual space topology of uniform convergence on bounded subsets
• Topological vector space – Vector space with a notion of nearness
• Topologies on the set of operators on a Hilbert space
• Uniform convergence
• Uniform space – Topological space with a notion of uniform properties
• Weak topology – Topology where convergence of points is defined by the convergence of their image under continuous linear functionals

References

1. Narici & Beckenstein 2011, pp. 371-423.
2. Jarchow 1981, pp. 43-55.
3. Narici & Beckenstein 2011, pp. 19-45.
4. Note that each set 𝒰(G, N) is a neighborhood of the origin for this topology, but it is not necessarily an open neighborhood of the origin.
5. Schaefer & Wolff 1999, pp. 79-88.
6. In practice, 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, 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.
7. Schaefer & Wolff 1999, p. 81.
8. Schaefer & Wolff 1999, p. 80.
9. Schaefer & Wolff 1999, p. 83.
10. Schaefer & Wolff 1999, p. 117.
11. Schaefer & Wolff 1999, p. 82.
12. Schaefer & Wolff 1999, p. 87.

Bibliography

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