Applications of the open mapping theorem for Banach spaces?

Does anybody know of any common/standard/famous practical applications of the open mapping theorem for Banach spaces?

Textbooks describe the theorem as a "cornerstone of functional analysis", and yet I have never come across a practical problem that is solved using it.

I do know that the open mapping theorem implies the inverse mapping theorem and the closed graph theorem. I would imagine the closed graph theorem to be of more direct applicability than the open mapping theorem itself. For instance, the closed graph theorem tells you that, in order to prove that a map $f : X \to Y$ between Banach spaces is continuous, you only have to prove that if $\lim_{n \to \infty}x_n = x$ and $\lim_{n \to \infty} f(x_n) = y$, then $y = f(x)$, i.e. it is okay to assume that the limit $\lim_{n \to \infty} f(x_n)$ exists. Still, it bothers me that I have never seen this technique applied to a concrete example that anyone cares about!

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.

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.

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.