---
tag: 3RCAY97W1E
kind: concept
name: Graph of an operator
msc: [47A05]
specializes: [0JBJYT69ZV]
sources:
  - url: https://en.wikipedia.org/wiki/Unbounded_operator
---

## Summary

The graph of an operator $T$ from $X$ to $Y$ is the set of pairs $(x, Tx)$ with $x$ in the domain of $T$.
It is a linear subspace of $X \times Y$, and it determines the operator completely, domain included.
Topological properties of the graph, such as being closed, replace boundedness in the theory of unbounded operators.

## Explanations

### L1 human

For a function of one real variable, the graph is the curve of points $(x, f(x))$ in the plane.
The graph of an operator is the same picture in the product space $X \times Y$.
For example, the graph of $x \mapsto 2x$ on $\R$ is the line $y = 2x$ in $\R^2$.
Because operators are linear, their graphs are always flat subspaces through the origin.