Unbounded operator

Builds on Normed vector space Linear subspace Linear map Continuous linear map

Definition

Let $X$ and $Y$ be normed spaces over $\R$ or $\C$ . An unbounded operator (or linear operator) $T$ from $X$ to $Y$ is a pair consisting of

a linear subspace $D(T)$ of $X$ , the domain of $T$ , and
a linear map $T \colon D(T) \to Y$ .

Two such operators $S$ and $T$ are equal when $D(S) = D(T)$ and $Sx = Tx$ for all $x \in D(T)$ . The operator $T$ is an extension of $S$ , written $S \subseteq T$ , when $D(S) \subseteq D(T)$ and $Tx = Sx$ for all $x \in D(S)$ .

Remarks

Example: on $X = Y = L^2(0, 1)$ , let $D(T)$ be the space of continuously differentiable functions on $[0, 1]$ , and let $Tf = f'$ . This is linear, so $T$ is an unbounded operator. It is not bounded: $f_n(x) = \sin(n \pi x)$ has $\norm{f_n} = 1/\sqrt{2}$ , while $\norm{T f_n} = n \pi / \sqrt{2}$ .

Non-example: the rule $f \mapsto f'$ is not an operator with domain all of $L^2(0, 1)$ , since many elements of $L^2(0, 1)$ have no derivative in $L^2(0, 1)$ .

Every continuous linear map $X \to Y$ is an unbounded operator with domain $X$ ; the word "unbounded" means "not necessarily bounded". Sums and products are defined on the natural domains: $D(S + T) = D(S) \cap D(T)$ and $D(ST) = \set{x \in D(T) : Tx \in D(S)}$ .

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Tag	`M9HEZ5X3FX`
URI	`m2://M9HEZ5X3FX`
Kind	definition
Writing	W1 stated
MSC	47A05, 47B25
Defines	Unbounded operator
Uses	Normed vector space, Linear subspace, Linear map, Continuous linear map
Changed	2 November 2026

Unbounded operator

Definition

Remarks

Reading path

Further reading

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