Applications of the open mapping theorem for Banach spaces?

A common example is the following:

If $\lVert \cdot \rVert_1$ and $\lVert \cdot \rVert_2$ are two norms in a vector space $X$ such that

• $\lVert \cdot \rVert_1\leq K\lVert \cdot \rVert_2$
• $(X,\lVert \cdot \rVert_1)$ and $(X,\lVert \cdot \rVert_2)$ are Banach

then $\lVert \cdot \rVert_1$ and $\lVert \cdot \rVert_2$ are equivalent.

The proof is easy: the linear operator $\text{Id}: (X,\lVert \cdot \rVert_2) \rightarrow (X,\lVert \cdot \rVert_1)$ is clearly surjective and continuous. So it must be open.

Here's one I just found myself: Let $f \in L^2(\mathbb R)$, and consider the map $L^2(\mathbb R) \to L^\infty (\mathbb R)$ defined by sending $g \in L^2(\mathbb R)$ to the convolution $f \star g \in L^\infty(\mathbb R)$. We can prove that this map is continuous.

To prove this, consider a sequence $g_n$ converging to $g$ in $L^2(\mathbb R)$. Assume that $f \star g_n $ converges to $h$ (say) in $L^\infty(\mathbb R)$. From the inequality $|f\star g_n (x) - f \star g(x)| \leq ||f||_{L^2} || g_n - g ||_{L^2}$ for all $x \in \mathbb R$, it follows that $h = f \star g$ in $L^\infty (\mathbb R)$. Then apply the closed mapping theorem.

But do let me know if you know of other examples - your help is much appreciated.

If $X$, $Y$ are Banach spaces and $F:X\to Y$ is a bounded linear transformation, which is one-to-one and onto, then its inverse is also bounded.