Closed linear operator

In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator.

The closed graph theorem says a closed linear operator between Banach spaces is continuous; thus is a bounded operator. Hence, a closed linear operator that is used in practice is typically not defined on a Banach space or some other complete spaces but is often defined on a dense subspace.

Definition
It is common in functional analysis to consider partial functions, which are functions defined on a subset of some space $$X.$$ A partial function $$f$$ is declared with the notation $$f : D \subseteq X \to Y,$$ which indicates that $$f$$ has prototype $$f : D \to Y$$ (that is, its domain is $$D$$ and its codomain is $$Y$$)

Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the graph of a partial function $$f$$ is the set $\operatorname{graph}{\!(f)} = \{(x, f(x)) : x \in \operatorname{dom} f\}.$ However, one exception to this is the definition of "closed graph". A function $$f : D \subseteq X \to Y$$ is said to have a closed graph if $$\operatorname{graph} f$$ is a closed subset of $$X \times Y$$ in the product topology; importantly, note that the product space is $$X \times Y$$ and  $$D \times Y = \operatorname{dom} f \times Y$$ as it was defined above for ordinary functions. In contrast, when $$f : D \to Y$$ is considered as an ordinary function (rather than as the partial function $$f : D \subseteq X \to Y$$), then "having a closed graph" would instead mean that $$\operatorname{graph} f$$ is a closed subset of $$D \times Y.$$ If $$\operatorname{graph} f$$ is a closed subset of $$X \times Y$$ then it is also a closed subset of $$\operatorname{dom} (f) \times Y$$ although the converse is not guaranteed in general.

Definition: If $X$ and $Y$ are topological vector spaces (TVSs) then we call a linear map $f : D(f) ⊆ X → Y$ a closed linear operator if its graph is closed in $X ×&thinsp;Y$.

Closable maps and closures
A linear operator $$f : D \subseteq X \to Y$$ is  in $$X \times Y$$ if there exists a $$E \subseteq X$$  containing $$D$$ and a function (resp. multifunction) $$F : E \to Y$$ whose graph is equal to the closure of the set $$\operatorname{graph} f$$ in $$X \times Y.$$ Such an $$F$$ is called a closure of $$f$$ in $$X \times Y$$, is denoted by $$\overline{f},$$ and necessarily extends $$f.$$

If $$f : D \subseteq X \to Y$$ is a closable linear operator then a ' or an ' of $$f$$ is a subset $$C \subseteq D$$ such that the closure in $$X \times Y$$ of the graph of the restriction $$f\big\vert_C : C \to Y$$ of $$f$$ to $$C$$ is equal to the closure of the graph of $$f$$ in $$X \times Y$$ (i.e. the closure of $$\operatorname{graph} f$$ in $$X \times Y$$ is equal to the closure of $$\operatorname{graph} f\big\vert_C$$ in $$X \times Y$$).

Examples
A bounded operator is a closed operator. Here are examples of closed operators that are not bounded.

  If $$(X, \tau)$$ is a Hausdorff TVS and $$\nu$$ is a vector topology on $$X$$ that is strictly finer than $$\tau,$$ then the identity map $$\operatorname{Id} : (X, \tau) \to (X, \nu)$$ a closed discontinuous linear operator.   Consider the derivative operator $$A = \frac{d}{d x}$$ where $$X = Y = C([a, b]).$$is the Banach space of all continuous functions on an interval $$[a, b].$$ If one takes its domain $$D(f)$$ to be $$C^1([a, b]),$$ then $$f$$ is a closed operator, which is not bounded. On the other hand, if $$D(f)$$ is the space $$C^\infty([a, b])$$ of smooth functions scalar valued functions then $$f$$ will no longer be closed, but it will be closable, with the closure being its extension defined on $$C^1([a, b]).$$  

Basic properties
The following properties are easily checked for a linear operator $f : D(f) ⊆ X → Y$ between Banach spaces:


 * If $A$ is closed then $A − λId_{D(f)}$ is closed where $λ$ is a scalar and $Id_{D(f)}$ is the identity function;
 * If $f$ is closed, then its kernel (or nullspace) is a closed vector subspace of $X$;
 * If $f$ is closed and injective then its inverse $f&thinsp;^{−1}$ is also closed;
 * A linear operator $f$ admits a closure if and only if for every $x ∈ X$ and every pair of sequences $x_{•} = (x_{i})∞ i=1$ and $y_{•} = (y_{i})∞ i=1$ in $D(f)$ both converging to $x$ in $X$, such that both $f(x_{•}) = (f(x_{i}))∞ i=1$ and $f(y_{•}) = (f(y_{i}))∞ i=1$ converge in $Y$, one has $lim_{i → ∞} fx_{i} = lim_{i → ∞} fy_{i}$.