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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!

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  • $\begingroup$ One good application is to proof that, starting with only genuinely twice continuously differentiable functions in $L^2(\mathbb R)$ or similar, there is simply no continuous linear operator on that Hilbert space extending it, whether one uses AxCh or anything else. $\endgroup$ – paul garrett Feb 23 '17 at 23:26
  • $\begingroup$ @paulgarrett Sorry could you please clarify - extending what exactly? Thanks in advance. $\endgroup$ – Kenny Wong Feb 23 '17 at 23:35
  • $\begingroup$ Oop, sorry, extending the operator "differentiation", or the symmetric operator "second derivative" (=one-dimensional Laplacian...) $\endgroup$ – paul garrett Feb 23 '17 at 23:38
  • $\begingroup$ Oh right, I see. $\endgroup$ – Kenny Wong Feb 23 '17 at 23:41
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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 $Id: (X,\lVert \cdot \rVert_2) \rightarrow (X,\lVert \cdot \rVert_1)$ is clearly surjective and continuous. So it must be open.

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  • $\begingroup$ That's really nice. I didn't know this was true. (I have wasted lots of time in my life checking $||.||_2 \leq k ||.||_1 \leq K ||.||_2$!) I suppose this result follows directly from the inverse mapping version of the theorem too. $\endgroup$ – Kenny Wong Feb 23 '17 at 23:31
  • $\begingroup$ Yes, it is true. But remember that this only applies when $X$ becomes a Banach spaces with both norms. $\endgroup$ – A. Salguero-Alarcón Feb 23 '17 at 23:37
  • $\begingroup$ Ah yes, good point. $\endgroup$ – Kenny Wong Feb 23 '17 at 23:39
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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.

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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.

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One of the corollaries from the open mapping theorem is quite important in the interpolation theory.

Corollary: There is a constant $M>0$ such that for every $y\in Y$ there is $x\in T^{-1}(y)$ satisfying $\|x\|_X \leq M\|y\|_Y$.

A sequence $\{z_k\}_{k=1}^\infty$ in the unit disk is interpolating if for every bounded sequence $\{w_k\}_{k=1}^\infty$ there is a function $f(x)\in H^\infty$ such that $f(z_k) = w_k,\; k=1,2,...$

On the other hand, $\{z_k\}_{k=1}^\infty$ is an interpolating sequence when the operator $E:H^\infty\to l^\infty$ defined by $E(f)=\{f(z_k)\}_{k=1}^\infty$ is surjective. So by the aforementioned corollary from the open mapping theorem we have $$ \|f\|_\infty \leq M \|\{w_k\}_{k=1}^\infty\|_\infty $$

The inequality is used to prove the Carleson condition on interpolating sequences in $H^\infty$ spaces.

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