Functional Analysis – Applications of the Open Mapping Theorem for Banach Spaces

functional-analysis

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!

Best Answer

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.

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