---
tag: VM7B38PA87
kind: concept
name: Closed operator
msc: [47A05, 47B25]
specializes: [KRJX5RA0PW]
sources:
  - url: https://en.wikipedia.org/wiki/Closed_linear_operator
same_as:
  wikidata: Q320370
  wikipedia: Closed linear operator
---

## Summary

A closed operator is an unbounded operator whose graph is a closed subset of the product space.
Closedness is the weak substitute for continuity that most differential operators have.
On Banach spaces, the closed graph theorem says that a closed operator defined on the whole space is bounded.
So interesting closed operators are usually defined on a proper subspace.

## Explanations

### L1 human

A continuous map sends convergent sequences to convergent sequences.
A closed operator only promises something weaker: if $x_n \to x$ and the outputs $T x_n$ also converge, then $x$ is in the domain and the limit of the outputs is $Tx$.
Differentiation is a typical example: if functions converge and their derivatives converge uniformly, then the limit is differentiable with the limit derivative.
So the operator never gives a wrong answer in the limit, even though it may blow up.