In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".^{[clarification needed]}

A closed operator that is used in practice is often densely defined.

Let ${\displaystyle X,Y}$ be topological vector spaces.

A densely defined linear operator ${\displaystyle T}$ from ${\displaystyle X}$ to ${\displaystyle Y}$ is a linear operator of type ${\displaystyle T:D(T)\to Y}$, such that ${\displaystyle D(T)}$ is a dense subset of ${\displaystyle X}$. In other words, ${\displaystyle T}$ is a partial function whose domain is dense in ${\displaystyle X}$.

Sometimes this is abbreviated as ${\displaystyle T:X\to Y}$ when the context makes it clear that ${\displaystyle T}$ might not be defined for all of ${\displaystyle X}$.

Closed Graph Theorem—If ${\displaystyle X,Y}$ are Hausdorff and metrizable, ${\displaystyle T:D(T)\to Y}$ is densely defined, with continuous inverse ${\displaystyle S:Y\to D(T)}$, then ${\displaystyle T}$ is closed. That is, the set ${\displaystyle \{(x,T(x)):x\in D(T)\}}$ is closed in the product topology of ${\displaystyle X\times Y}$.

Take any net ${\displaystyle (x_{\alpha })}$ in ${\displaystyle D(T)}$ with ${\displaystyle Tx_{\alpha }\to y}$ in ${\displaystyle Y}$. By continuity of ${\displaystyle S}$, ${\displaystyle x_{\alpha }=S(Tx_{\alpha })\to \ S(y)}$. Hence there exists some ${\displaystyle x\in D(T)}$ such that ${\displaystyle x_{\alpha }\to x}$, and ${\displaystyle Tx=T(S(y))=y}$.

The Hausdorff property ensures sequential convergence is unique. The metrizability property ensures that sequentially closed sets are closed. In functional analysis, these conditions typically hold, as most spaces under consideration are Fréchet space, or stronger than Fréchet. In particular, Banach spaces are Fréchet.