Open Mapping and Closed Graph Theorems - Examples and Constructions

The Bounded Inverse Theorem is an immediate corollary of the Open Mapping Theorem with far-reaching consequences for the study of operator equations and spectral theory.

TheoremBounded Inverse Theorem

Let $X$ and $Y$ be Banach spaces and $T : X \to Y$ a bounded linear operator. If $T$ is bijective, then $T^{-1} : Y \to X$ is also bounded.

This theorem eliminates the need to verify separately that the inverse of a bijective bounded operator is continuous—it's automatic in Banach spaces!

Proof

Since $T$ is bijective and bounded, by the Open Mapping Theorem, $T$ is an open mapping. For any open ball $B(0, r) \subset X$ , the image $T(B(0,r))$ is open in $Y$ and contains $0$ . Therefore, there exists $\delta > 0$ such that $B(0, \delta) \subset T(B(0, r))$ .

For any $y \in Y$ with $\|y\| \leq \delta$ , we have $y = T(x)$ for some $x \in B(0, r)$ , so $\|T^{-1}(y)\| = \|x\| < r$ . Thus $\|T^{-1}(y)\| \leq \frac{r}{\delta} \|y\|$ proving that $T^{-1}$ is bounded with $\|T^{-1}\| \leq r/\delta$ .

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ExampleApplications

Fredholm Alternative: For compact perturbations $T = I - K$ where $K$ is compact, injectivity implies surjectivity
Spectral Theory: If $(T - \lambda I)$ is bijective, then $\lambda$ is in the resolvent set and $\|(T - \lambda I)^{-1}\|$ is finite
Isomorphisms: Two Banach spaces are isomorphic if and only if there exists a bijective bounded linear operator between them
Change of Norms: If two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a vector space $X$ make it complete, and if the identity map $id : (X, \|\cdot\|_1) \to (X, \|\cdot\|_2)$ is bounded and bijective, then the norms are equivalent

TheoremContinuous Inverse Principle

Let $X$ and $Y$ be Banach spaces. A bounded linear operator $T : X \to Y$ is an isomorphism onto its range if and only if there exists $c > 0$ such that $\|T(x)\| \geq c \|x\|$ for all $x \in X$ .

Remark

The Bounded Inverse Theorem dramatically simplifies many proofs in functional analysis. Without it, one would need to explicitly estimate $\|T^{-1}\|$ in terms of $\|T\|$ , which can be extremely difficult. The theorem says this is unnecessary—boundedness of the inverse is automatic from bijectivity.

This result is fundamental to solving operator equations $T(x) = y$ : if $T$ is bijective, the solution operator $T^{-1}$ is automatically continuous, ensuring stability with respect to perturbations in the data $y$ .