Open Mapping and Closed Graph Theorems - Core Definitions

The Open Mapping Theorem and Closed Graph Theorem are among the most powerful tools in functional analysis, providing automatic continuity and surjectivity results for linear operators between Banach spaces.

DefinitionOpen Mapping

Let $X$ and $Y$ be topological spaces. A function $T : X \to Y$ is an open mapping if for every open set $U \subset X$ , the image $T(U)$ is open in $Y$ .

Equivalently, $T$ is open if it maps neighborhoods of points to neighborhoods of their images.

In general, continuous functions need not be open (consider $x \mapsto x^2$ from $\mathbb{R}$ to $\mathbb{R}$ ). However, for linear operators between Banach spaces, surjectivity and continuity together imply openness.

TheoremOpen Mapping Theorem

Let $X$ and $Y$ be Banach spaces and $T : X \to Y$ a bounded linear operator. If $T$ is surjective, then $T$ is an open mapping.

This theorem is remarkable: the algebraic property (surjectivity) combined with continuity automatically implies the topological property (openness). No additional conditions are needed.

ExampleConsequences

Bounded Inverse: If $T : X \to Y$ is a bijective bounded linear operator between Banach spaces, then $T^{-1}$ is automatically bounded
Isomorphism: Bijective bounded linear operators between Banach spaces are isomorphisms (algebraic and topological)
Automatic Continuity: Many "natural" operators are automatically continuous without explicit verification

DefinitionGraph of an Operator

Let $X$ and $Y$ be normed spaces and $T : X \to Y$ a linear operator (not necessarily bounded). The graph of $T$ is $\Gamma(T) = \{(x, T(x)) : x \in X\} \subset X \times Y$

The operator $T$ is said to have a closed graph if $\Gamma(T)$ is closed in $X \times Y$ (with the product topology).

Having a closed graph means: if $x_n \to x$ in $X$ and $T(x_n) \to y$ in $Y$ , then $y = T(x)$ . This is a weaker condition than continuity.

Remark

Every bounded linear operator has a closed graph (by continuity). The Closed Graph Theorem provides a converse for operators between Banach spaces: a closed graph implies boundedness.

These theorems rely critically on the completeness of both spaces. They fail spectacularly for incomplete spaces, showing once again the fundamental importance of Banach spaces in functional analysis.