Corollaries to the Open Mapping Theorem

# Corollaries to the Open Mapping Theorem

Recall from The Open Mapping Theorem page that if $X$ and $Y$ are Banach spaces and if $T : X \to Y$ is a bounded linear operator then the range $T(X)$ is closed if and only if $T$ is an open map.

We will now present some corollaries to the open mapping theorem.

Corollary 1 tells us that if $T$ is a bijective bounded linear operator from a Banach space $X$ to a Banach space $Y$ then the inverse $T^{-1}$ is also bounded.

 Corollary 1: Let $X$ and $Y$ be Banach spaces and let $T : X \to Y$ be a bounded linear operator. If $T$ is bijective then $T^{-1}$ is a bounded linear operator.
• Proof: Since $T$ is bijective, $T(X) = Y$, which is closed. So by the open mapping theorem we have that $T$ is an open map. But again since $T$ is bijective, $T$ being open implies that $T^{-1}$ is a bounded linear operator. $\blacksquare$

Corollary 2 gives us a quicker way to determine if two norms on a space are equivalent or not - provided that $X$ equipped with both norms forms a Banach space.

 Corollary 2: Let $(X, \| \cdot \|_1)$ and $(X, \| \cdot \|_2)$ be Banach spaces. If there is an $M > 0$ for which $\| x \|_2 \leq M \| x \|_1$ for every $x \in X$ then $\| \cdot \|_1$ and $\| \cdot \|_2$ are equivalent norms.
• Proof: Let $i : (X, \| \cdot \|_1) \to (X, \| \cdot \|_2)$ be the identity map defined for all $x \in X$ by $i(x) = x$. Then for all $x \in X$ we have that:
(1)
\begin{align} \quad \| i(x) \|_2 \leq M \| x \|_1 \end{align}
• So $i$ is bounded. Since $(X, \| \cdot \|_1)$ and $(X, \| \cdot \|_2)$ are Banach spaces and since $i$ is bijective, by corollary $1$ we have that $i^{-1} : (X, \| \cdot \|_2) \to (X, \| \cdot \|_1)$ is a bounded linear operator. So there exists an $m > 0$ such that:
(2)
\begin{align} \quad \| i^{-1}(x) \|_1 \leq m \| x \|_2 \end{align}
• For all $x \in X$. Hence for all $x \in X$:
(3)
\begin{align} \quad \frac{1}{m} \| x \|_1 \leq \| x \|_2 \leq M \| x \|_1 \end{align}
• So $\| \cdot \|_1$ and $\| \cdot \|_2$ are equivalent norms. $\blacksquare$