Closed graph property

:Definition and notation: The graph of a function {{math|f : X → Y}} is the set

::{{math|1=Gr f := { (x, f(x)) : x ∈ X } = { (x, y) ∈ X × Y : y = f(x) }}}.

:Notation: If {{mvar|Y}} is a set then the power set of {{mvar|Y}}, which is the set of all subsets of {{mvar|Y}}, is denoted by {{math|2^Y}} or {{math|𝒫(Y)}}.

:Definition: If {{mvar|X}} and {{mvar|Y}} are sets, a set-valued function in {{mvar|Y}} on {{mvar|X}} (also called a {{mvar|Y}}-valued multifunction on {{mvar|X}}) is a function {{math|F : X → 2^Y}} with domain {{mvar|X}} that is valued in {{math|2^Y}}. That is, {{mvar|F}} is a function on {{mvar|X}} such that for every {{math|x ∈ X}}, {{math|F(x)}} is a subset of {{mvar|Y}}.

:* Some authors call a function {{math|F : X → 2^Y}} a set-valued function only if it satisfies the additional requirement that {{math|F(x)}} is not empty for every {{math|x ∈ X}}; this article does not require this.

:Definition and notation: If {{math|F : X → 2^Y}} is a set-valued function in a set {{mvar|Y}} then the graph of {{mvar|F}} is the set

::{{math|1=Gr F := { (x, y) ∈ X × Y : y ∈ F(x) }}}.

:Definition: A function {{math|f : X → Y}} can be canonically identified with the set-valued function {{math|F : X → 2^Y}} defined by {{math|1=F(x) := { f(x) } }} for every {{math|x ∈ X}}, where {{mvar|F}} is called the canonical set-valued function induced by (or associated with) {{mvar|f}}.

:*Note that in this case, {{math|1=Gr f = Gr F}}.

We give the more general definition of when a {{mvar|Y}}-valued function or set-valued function defined on a subset {{mvar|S}} of {{mvar|X}} has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace {{mvar|S}} of a topological vector space {{mvar|X}} (and not necessarily defined on all of {{mvar|X}}).

This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.

:Assumptions: Throughout, {{mvar|X}} and {{mvar|Y}} are topological spaces, {{math|S ⊆ X}}, and {{mvar|f}} is a {{mvar|Y}}-valued function or set-valued function on {{mvar|S}} (i.e. {{math|f : S → Y}} or {{math|f : S → 2^Y}}). {{math|X × Y}} will always be endowed with the product topology.

:Definition:{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} We say that {{mvar|f}}  has a closed graph in {{math|X × Y}} if the graph of {{mvar|f}}, {{math|Gr f}}, is a closed subset of {{math|X × Y}} when {{math|X × Y}} is endowed with the product topology. If {{math|1=S = X}} or if {{mvar|X}} is clear from context then we may omit writing "in {{math|X × Y}}"

Note that we may define an open graph, a sequentially closed graph, and a sequentially open graph in similar ways.

:Observation: If {{math|g : S → Y}} is a function and {{mvar|G}} is the canonical set-valued function induced by {{mvar|g}}  (i.e. {{math|G : S → 2^Y}} is defined by {{math|1=G(s) := { g(s) } }} for every {{math|s ∈ S}}) then since {{math|1=Gr g = Gr G}}, {{mvar|g}} has a closed (resp. sequentially closed, open, sequentially open) graph in {{math|X × Y}} if and only if the same is true of {{mvar|G}}.

:Definition: We say that the function (resp. set-valued function) {{mvar|f}} is closable in {{math|X × Y}} if there exists a subset {{math|D ⊆ X}} containing {{mvar|S}} and a function (resp. set-valued function) {{math|F : D → Y}} whose graph is equal to the closure of the set {{math|Gr f}} in {{math|X × Y}}. Such an {{mvar|F}} is called a closure of {{mvar|f}} in {{math|X × Y}}, is denoted by {{math|{{overline|f}}}}, and necessarily extends {{mvar|f}}.

:*Additional assumptions for linear maps: If in addition, {{mvar|S}}, {{mvar|X}}, and {{mvar|Y}} are topological vector spaces and {{math|f : S → Y}} is a linear map then to call {{mvar|f}} closable we also require that the set {{mvar|D}} be a vector subspace of {{mvar|X}} and the closure of {{mvar|f}} be a linear map.

:Definition: If {{mvar|f}} is closable on {{mvar|S}} then a core or essential domain of {{mvar|f}} is a subset {{math|D ⊆ S}} such that the closure in {{math|X × Y}} of the graph of the restriction {{math|f {{big|{{!}}}}_D : D → Y}} of {{mvar|f}} to {{mvar|D}} is equal to the closure of the graph of {{mvar|f}} in {{math|X × Y}} (i.e. the closure of {{math|Gr f}} in {{math|X × Y}} is equal to the closure of {{math|Gr f {{big|{{!}}}}_D}} in {{math|X × Y}}).

:Definition and notation: When we write {{math|f : D(f) ⊆ X → Y}} then we mean that {{mvar|f}} is a {{mvar|Y}}-valued function with domain {{math|D(f)}} where {{math|D(f) ⊆ X}}. If we say that {{math|f : D(f) ⊆ X → Y}} is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) then we mean that the graph of {{mvar|f}} is closed (resp. sequentially closed) in {{math|X × Y}} (rather than in {{math|D(f) × Y}}).

When reading literature in functional analysis, if {{math|f : X → Y}} is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then "{{mvar|f}} is closed" will almost always means the following:

:Definition: A map {{math|f : X → Y}} is called closed if its graph is closed in {{math|X × Y}}. In particular, the term "closed linear operator" will almost certainly refer to a linear map whose graph is closed.

Otherwise, especially in literature about point-set topology, "{{mvar|f}} is closed" may instead mean the following:

:Definition: A map {{math|f : X → Y}} between topological spaces is called a closed map if the image of a closed subset of {{mvar|X}} is a closed subset of {{mvar|Y}}.

These two definitions of "closed map" are not equivalent.

If it is unclear, then it is recommended that a reader check how "closed map" is defined by the literature they are reading.

Throughout, let {{mvar|X}} and {{mvar|Y}} be topological spaces.

;Function with a closed graph

If {{math|f : X → Y}} is a function then the following are equivalent:

1. {{mvar|f}}  has a closed graph (in {{math|X × Y}});
2. (definition) the graph of {{mvar|f}}, {{math|Gr f}}, is a closed subset of {{math|X × Y}};
3. for every {{math|x ∈ X}} and net {{math|1=x_• = (x_i)_{i ∈ I}}} in {{mvar|X}} such that {{math|x_• → x}} in {{mvar|X}}, if {{math|y ∈ Y}} is such that the net {{math|1=f(x_•) := (f(x_i))_{i ∈ I} → y}} in {{mvar|Y}} then {{math|1=y = f(x)}};{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}
4. * Compare this to the definition of continuity in terms of nets, which recall is the following: for every {{math|x ∈ X}} and net {{math|1=x_• = (x_i)_{i ∈ I}}} in {{mvar|X}} such that {{math|x_• → x}} in {{mvar|X}}, {{math|f(x_•) → f(x)}} in {{mvar|Y}}.
5. * Thus to show that the function {{mvar|f}} has a closed graph we may assume that {{math|f(x_•)}} converges in {{mvar|Y}} to some {{math|y ∈ Y}} (and then show that {{math|1=y = f(x)}}) while to show that {{mvar|f}} is continuous we may not assume that {{math|f(x_•)}} converges in {{mvar|Y}} to some {{math|y ∈ Y}} and we must instead prove that this is true (and moreover, we must more specifically prove that {{math|f(x_•)}} converges to {{math|f(x)}} in {{mvar|Y}}).

and if {{mvar|Y}} is a Hausdorff space that is compact, then we may add to this list:

and if both {{mvar|X}} and {{mvar|Y}} are first-countable spaces then we may add to this list:

;Function with a sequentially closed graph

If {{math|f : X → Y}} is a function then the following are equivalent:

1. {{mvar|f}}  has a sequentially closed graph (in {{math|X × Y}});
2. (definition) the graph of {{mvar|f}} is a sequentially closed subset of {{math|X × Y}};
3. for every {{math|x ∈ X}} and sequence {{math|1=x_• = (x_i){{su|p=∞|b=i=1}}}} in {{mvar|X}} such that {{math|x_• → x}} in {{mvar|X}}, if {{math|y ∈ Y}} is such that the net {{math|1=f(x_•) := (f(x_i)){{su|p=∞|b=i=1}} → y}} in {{mvar|Y}} then {{math|1=y = f(x)}};{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}

;set-valued function with a closed graph

If {{math|F : X → 2^Y}} is a set-valued function between topological spaces {{mvar|X}} and {{mvar|Y}} then the following are equivalent:

1. {{mvar|F}}  has a closed graph (in {{math|X × Y}});
2. (definition) the graph of {{mvar|F}} is a closed subset of {{math|X × Y}};

and if {{mvar|Y}} is compact and Hausdorff then we may add to this list:

and if both {{mvar|X}} and {{mvar|Y}} are metrizable spaces then we may add to this list:

Throughout, let $X$ and $Y$ be topological spaces and $X\; \backslash times\; Y$ is endowed with the product topology.

If $f\; :\; X\; \backslash to\; Y$ is a function then it is said to have a {{em|closed graph}} if it satisfies any of the following are equivalent conditions:

1. (Definition): The graph $\backslash operatorname\{graph\}\; f$ of $f$ is a closed subset of $X\; \backslash times\; Y.$
2. For every $x\; \backslash in\; X$ and net $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ in $X$ such that $x\_\{\backslash bull\}\; \backslash to\; x$ in $X,$ if $y\; \backslash in\; Y$ is such that the net $f\backslash left(x\_\{\backslash bull\}\backslash right)\; =\; \backslash left(f\backslash left(x\_i\backslash right)\backslash right)\_\{i\; \backslash in\; I\}\; \backslash to\; y$ in $Y$ then $y\; =\; f(x).${{sfn|Narici|Beckenstein|2011|pp=459-483}}
   • Compare this to the definition of continuity in terms of nets, which recall is the following: for every $x\; \backslash in\; X$ and net $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ in $X$ such that $x\_\{\backslash bull\}\; \backslash to\; x$ in $X,$ $f\backslash left(x\_\{\backslash bull\}\backslash right)\; \backslash to\; f(x)$ in $Y.$
   • Thus to show that the function $f$ has a closed graph, it may be assumed that $f\backslash left(x\_\{\backslash bull\}\backslash right)$ converges in $Y$ to some $y\; \backslash in\; Y$ (and then show that $y\; =\; f(x)$) while to show that $f$ is continuous, it may not be assumed that $f\backslash left(x\_\{\backslash bull\}\backslash right)$ converges in $Y$ to some $y\; \backslash in\; Y$ and instead, it must be proven that this is true (and moreover, it must more specifically be proven that $f\backslash left(x\_\{\backslash bull\}\backslash right)$ converges to $f(x)$ in $Y$).

and if $Y$ is a Hausdorff compact space then we may add to this list:

3. $f$ is continuous.{{sfn|Munkres|2000|p=171}}

and if both $X$ and $Y$ are first-countable spaces then we may add to this list:

4. $f$ has a sequentially closed graph in $X\; \backslash times\; Y.$

Function with a sequentially closed graph

If $f\; :\; X\; \backslash to\; Y$ is a function then the following are equivalent:

1. $f$ has a sequentially closed graph in $X\; \backslash times\; Y.$
2. Definition: the graph of $f$ is a sequentially closed subset of $X\; \backslash times\; Y.$
3. For every $x\; \backslash in\; X$ and sequence $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X$ such that $x\_\{\backslash bull\}\; \backslash to\; x$ in $X,$ if $y\; \backslash in\; Y$ is such that the net $f\backslash left(x\_\{\backslash bull\}\backslash right)\; :=\; \backslash left(f\backslash left(x\_i\backslash right)\backslash right)\_\{i=1\}^\{\backslash infty\}\; \backslash to\; y$ in $Y$ then $y\; =\; f(x).${{sfn|Narici|Beckenstein|2011|pp=459-483}}

• If {{math|f : X → Y}} is a continuous function between topological spaces and if {{mvar|Y}} is Hausdorff then {{mvar|f}}  has a closed graph in {{math|X × Y}}.{{sfn | Narici | Beckenstein | 2011 | pp=459-483}} However, if {{math|f}} is a function between Hausdorff topological spaces, then it is possible for {{mvar|f}}  to have a closed graph in {{math|X × Y}} but not be continuous.

{{Main|Closed graph theorem}}

Conditions that guarantee that a function with a closed graph is necessarily continuous are called closed graph theorems.

Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous.

• If {{math|f : X → Y}} is a function between topological spaces whose graph is closed in {{math|X × Y}} and if {{mvar|Y}} is a compact space then {{math|f : X → Y}} is continuous.{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}

For examples in functional analysis, see continuous linear operator.

• Let {{mvar|X}} denote the real numbers {{math|ℝ}} with the usual Euclidean topology and let {{mvar|Y}} denote {{math|ℝ}} with the indiscrete topology (where note that {{mvar|Y}} is not Hausdorff and that every function valued in {{mvar|Y}} is continuous). Let {{math|f : X → Y}} be defined by {{math|1=f(0) = 1}} and {{math|1=f(x) = 0}} for all {{math|x ≠ 0}}. Then {{math|f : X → Y}} is continuous but its graph is not closed in {{math|X × Y}}.{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}
• If {{mvar|X}} is any space then the identity map {{math|Id : X → X}} is continuous but its graph, which is the diagonal {{math|1=Gr Id := { (x, x) : x ∈ X }}}, is closed in {{math|X × X}} if and only if {{mvar|X}} is Hausdorff.Rudin p.50 In particular, if {{mvar|X}} is not Hausdorff then {{math|Id : X → X}} is continuous but not closed.
• If {{math|f : X → Y}} is a continuous map whose graph is not closed then {{mvar|Y}} is not a Hausdorff space.

• Let {{mvar|X}} and {{mvar|Y}} both denote the real numbers {{math|ℝ}} with the usual Euclidean topology. Let {{math|f : X → Y}} be defined by {{math|1=f(0) = 0}} and {{math|1=f(x) = {{sfrac|1|x}}}} for all {{math|x ≠ 0}}. Then {{math|f : X → Y}} has a closed graph (and a sequentially closed graph) in {{math|1=X × Y = ℝ²}} but it is not continuous (since it has a discontinuity at {{math|1=x = 0}}).{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}
• Let {{mvar|X}} denote the real numbers {{math|ℝ}} with the usual Euclidean topology, let {{mvar|Y}} denote {{math|ℝ}} with the discrete topology, and let {{math|Id : X → Y}} be the identity map (i.e. {{math|1=Id(x) := x}} for every {{math|x ∈ X}}). Then {{math|Id : X → Y}} is a linear map whose graph is closed in {{math|1=X × Y}} but it is clearly not continuous (since singleton sets are open in {{mvar|Y}} but not in {{mvar|X}}).{{sfn | Narici | Beckenstein | 2011 | pp=459-483}}
• Let {{math|(X, 𝜏)}} be a Hausdorff TVS and let {{math|𝜐}} be a vector topology on {{mvar|X}} that is strictly finer than {{math|𝜏}}. Then the identity map {{math|Id : (X, 𝜏) → (X, 𝜐)}} is a closed discontinuous linear operator.{{sfn | Narici | Beckenstein | 2011 | p=480}}

• {{annotated link|Almost open linear map}}
• {{annotated link|Closed graph theorem}}
• {{annotated link|Closed graph theorem (functional analysis)}}
• {{annotated link|Kakutani fixed-point theorem}}
• {{annotated link|Open mapping theorem (functional analysis)}}
• {{annotated link|Webbed space}}
• Graph continuous function

{{reflist}}

• {{Köthe Topological Vector Spaces I}}
• {{Kriegl Michor The Convenient Setting of Global Analysis}}
• {{Munkres Topology|edition=2}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Rudin Walter Functional Analysis|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Swartz An Introduction to Functional Analysis}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wilansky Modern Methods in Topological Vector Spaces}}

Category:Functional analysis