Closed linear operator

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

The closed graph theorem states that a linear operator

f

:

X

→

Y

{\displaystyle f:X\to Y}

between Banach spaces with full domain

X

{\displaystyle X}

is a closed operator if and only if it is a bounded operator. In practice, many operators are unbounded, but it is still desirable to make them have a closed graph. Hence, they cannot be defined on all of

X

{\displaystyle X}

. To stay useful, they are instead defined on a proper but dense subspace, which still allows approximating any vector and keeps key tools (closures, adjoints, spectral theory) available.

Definition

It is common in functional analysis to consider partial functions, which are functions defined on a subset of some space

X

.

{\displaystyle X.}

A partial function

f

{\displaystyle f}

is declared with the notation

f

:

D

⊆

X

→

Y

,

{\displaystyle f:D\subseteq X\to Y,}

which indicates that

f

{\displaystyle f}

has prototype

f

:

D

→

Y

{\displaystyle f:D\to Y}

(that is, its domain is

D

{\displaystyle D}

and its codomain is

Y

{\displaystyle 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

{\displaystyle f}

is the set

graph

⁡

(

f

)

=

{

(

x

,

f

(

x

)

)

:

x

∈

dom

⁡

f

}

.

{\displaystyle \operatorname {graph} {\!(f)}=\{(x,f(x)):x\in \operatorname {dom} f\}.}

However, one exception to this is the definition of "closed graph".