Open Mapping and Closed Graph Theorems - Main Theorem

Banach's Isomorphism Theorem combines the Open Mapping and Bounded Inverse Theorems into a comprehensive statement about when two Banach spaces are topologically equivalent.

TheoremBanach's Isomorphism Theorem

Let $X$ and $Y$ be Banach spaces and $T : X \to Y$ a bounded linear operator. The following are equivalent:

$T$ is an isomorphism (bijective with bounded inverse)
$T$ is bijective
$T$ is injective with closed range equal to $Y$
There exist constants $c, C > 0$ such that $c\|x\| \leq \|T(x)\| \leq C\|x\|$ for all $x \in X$

This theorem shows that for bounded linear operators between Banach spaces, algebraic bijectivity automatically implies topological bijectivity (homeomorphism).

Proof

$(1) \Rightarrow (2)$ : Obvious.

$(2) \Rightarrow (1)$ : By the Bounded Inverse Theorem.

$(2) \Rightarrow (3)$ : If $T$ is bijective, then $T$ is injective and $\text{Range}(T) = Y$ .

$(3) \Rightarrow (2)$ : If $T$ is injective with range equal to $Y$ , then $T$ is bijective.

$(1) \Rightarrow (4)$ : If $T^{-1}$ is bounded, then $\|x\| = \|T^{-1}(T(x))\| \leq \|T^{-1}\| \|T(x)\|$ , so $\|T(x)\| \geq \|T^{-1}\|^{-1} \|x\|$ . Set $c = \|T^{-1}\|^{-1}$ and $C = \|T\|$ .

$(4) \Rightarrow (1)$ : The lower bound $\|T(x)\| \geq c\|x\|$ implies $T$ is injective and has closed range. The upper bound $\|T(x)\| \leq C\|x\|$ says $T$ is bounded. If $T$ is surjective, then $T^{-1}$ exists and $\|T^{-1}(y)\| \leq c^{-1}\|y\|$ .

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ExampleApplications

Equivalent Formulations: Different formulations of PDE problems (strong, weak, variational) define equivalent operator equations when appropriate function spaces are chosen
Perturbation Theory: If $T : X \to Y$ is an isomorphism and $S : X \to Y$ satisfies $\|S\| < \|T^{-1}\|^{-1}$ , then $T + S$ is also an isomorphism
Continuous Dependence: Solutions to $T(x) = y$ depend continuously on $y$ when $T$ is an isomorphism
Stability: Numerical methods converge if the discrete operators are uniformly isomorphic to the continuous operator

Remark

The equivalence $(2) \Leftrightarrow (1)$ is the essence of the Bounded Inverse Theorem and relies critically on the Open Mapping Theorem. This fails for operators between incomplete normed spaces, where bijective bounded operators can have unbounded inverses.

This theorem is central to the study of well-posed problems in PDEs, where existence, uniqueness, and continuous dependence on data are all captured by the isomorphism property.