---
tag: BCJHNHKQFG
kind: concept
name: Densely defined operator
msc: [47A05, 47B25]
specializes: [KRJX5RA0PW]
sources:
  - url: https://en.wikipedia.org/wiki/Densely_defined_operator
same_as:
  wikidata: Q5259314
  wikipedia: Densely defined operator
---

## Summary

A densely defined operator is an unbounded operator whose domain is a dense subspace of the space.
Density of the domain is the condition under which the adjoint of an operator on a Hilbert space is well defined.
Differential operators on $L^2$ spaces, defined on smooth functions, are the standard examples.

## Explanations

### L1 human

An unbounded operator often cannot act on every vector, but it is good if it acts on vectors that come arbitrarily close to any vector.
That is what a dense domain gives.
For example, differentiation acts on polynomials, and every continuous function on $[0, 1]$ is a uniform limit of polynomials.
A dense domain also makes the operator determined by enough data to define its adjoint.