The Open Mapping Theorem
The open mapping theorem (or the Banach-Schauder theorem, if you prefer) is an incredibly important, relatively straightforward and digestible result in functional analysis which plays a crucial role in a large variety of other interesting theorems. As an exercise, we’ll prove the open mapping theorem here in the standard fashion. This will hopefully serve as a useful reference for later posts about metric regularity and linear openness, which serve as a measuring device to quantify the degree to which a map is open.
Stating the Open Mapping Theorem
The open mapping theorem may be stated as follows:
Restating this another way (with denoting the open unit ball), the theorem reads:
At the heart of the open mapping theorem is the notion that there is a link between precision and isomorphism among complete normed spaces (though the theorem holds under the even weaker assumption of local convexity as well): if the equation has at least one solution for any , then either is an isomorphism and is isomorphic to , or is a quotient map and is isomorphic to a quotient of . Taken another way, the open mapping theorem links precision and estimation — at least in the sense that if has a solution for any , then there exists a constant such that . More specifically (and this keys us in to precisely how open mappings relate to metric regularity), We say that the value , as defined here, is the Banach constant of , and will prove useful in the future.
We will prove the open mapping theorem in detail to highlight precisely the epsilon-delta argument used, which will hopefully allow us to see how weakening of linear openness may be better understood.
Proving the Open Mapping Theorem
The modern definition of a Baire space is often given in one of the four following equivalent forms:
That is, Baire spaces preserve density under countable intersections (notice that open need not be specified, as a dense closed set is necessarily the entire space).
That is, Baire spaces preserve boundary sets under countable unions (notice that closed need not be specified, as an open set with empty interior is necessarily the empty set).
The historical definition of a Baire space involves Baire’s notion of categories (not to be confused with the categories of category theory), but is also equivalent.
This leads to Baire’s original definition, which is as follows:
That is to say, open sets in Baire spaces are, in a sense, suitably ‘substantial’ or ‘large’ — at least, insofar as Baire spaces have no open sets which are meager.
Correspondingly, we also can find one further historical definition of a Baire space using this notion:
Seeing the straight-up crazy number of different definitions of a Baire space, one might wonder why these spaces deserve so much fuss. To this end, let us observe that Baire spaces enjoy a combinatorial property akin to the pigeonhole principle, which is (very obviously) equivalent to their definition.
So, Baire spaces are useful in some contexts because they allow us to — in a somewhat combinatorial fashion — extract data about the existence of a subset satisfying a certain property among a collection of subsets by looking at the properties of a larger set which contains them. Determining if a topological space is a Baire space allows us to utilize arguments of this form, so we frequently are interested in conditions dictating whether a topological space does indeed have this desirable property. In particular, we could consider this question in the context of a complete metric space, which leads us to the so-called Baire Category Theorem.
Let be a complete metric space. Using definition the first definition of a Baire space, we seek to show that a countable intersection of dense subsets is dense. To that end, let be a countable collection of dense subsets of . As a subset is dense if and only if every nonempty open subset intersects it, it is then sufficient to show that any nonempty open subset has a point which lies in the intersection of with each .
Proceeding in this fashion, observe that since is dense, . Thus, there exists and real constant such that .
Now, observe that, as is dense, , and we may find a point in the intersection and positive radius such that . Continuing recursively, we find a pair of sequences and such that and . Thus, we have a nested sequence of closed and bounded subsets , which are correspondingly compact by the Heine-Borel theorem. Applying Cantor’s intersection theorem, this then yields a fixed point
Moreover, as the sequence is Cauchy and is complete, .
Therefore, we may conclude that the intersection of a countable number of dense open subsets of a complete metric space is dense, and that every complete metric space is correspondingly a Baire space.
We will use this result in a central way to prove the open mapping theorem. However, we first need three lemmas.
Let be a Banach space and an arbitrary sequence in such that converges. It then follows that the partial sums of the series are a Cauchy sequence, and by the completeness of , the series converges to an element of .
Let be a normed space and suppose that every absolutely convergent series in converges in . We must now show that every Cauchy sequence in converges. To that end, let be an arbitrary Cauchy sequence in , and let be a subsequence of such that . It then follows that converges, and by our assumption, that converges as well to some . Observe that
As this series converges, it then follows that for some . Thus, we have shown that is a convergent subsequence of the Cauchy sequence and, correspondingly, we may conclude that converges in . Being that was arbitrary, it then follows that is complete and, thus, a Banach space.
Let be a Banach space, a normed space, and . Further, suppose we have a constant such that . Noting that scaling is a homeomorphism, without loss of generality we take by alternatively considering .
Having done this, let us first choose an arbitrary , and choose such that (that is, ). Then, choose by , observing that . We will demonstrate that , which correspondingly gives that .
To do so, we find a sequence such that
That is, a sequence which converges to and has the difference of successive terms in successively smaller contractions of .
As , is the limit of a sequence where . Then, it follows that for all , there exists such that for all . Following this, set . Now, let for and (which does indeed work, as ). Notice that this allows us to yield the following observations:
Thus, as , we then have that , and our desired properties have been fulfilled.
Following this, we find a convergent sequence in which also converges to under to demonstrate that . That is, we seek a sequence such that and . To do this, let us first notice that, for , we have , as . Setting , clearly by linearity, and the following then holds for :
Thus, . Following this, we may now also notice that
so the series is absolutely convergent, and correspondingly by the previous lemma, we then have that by the completeness of the space.
Moreover, note , so . Thus, by the linearity of , we then have that
and, correspondingly, . Hence, it follows that
As such, we have demonstrated that, if , then as well. As was chosen arbitrarily, this yields that as desired.
Let and be -vector spaces, be a bounded linear map, and be convex in . We seek to show that, for all , for all .
To that end, let and be arbitrary. Then, and for some . Moreover, by the convexity of , we then have that . But, by the linearity of , we yield the following inclusion
Thus, is convex as well.
Now, we use these results to prove the open mapping theorem, which we will recall is stated as follows:
Let and be Banach spaces and a surjective bounded linear operator. If is the trivial space, then we are done. Suppose is not the trivial space. By the linearity of the spaces, it is sufficient to show that maps to a neighborhood of the origin of .
First, let us note that . Correspondingly, by the surjectivity and linearity of , we then have that
As and are Banach spaces, they are also Baire spaces. Correspondingly, as the whole space has nonempty interior, there exists such that . Thus, there then must exist and such that .
Moreover, observe that if , then as well by the linearity of . Thus, , and as the image of a convex set under a bounded linear map is convex by the third lemma above, we then have that
By the second lemma, we then may conclude that . Therefore, as we have shown that maps to a neighborhood of the origin of , it follows that is then an open map.
This post draws on a large number of resources that I’ve encountered at various points over the last few years, most of which I didn’t write down at the time. If you recognize any of the proofs given above, I’d love to know where you’ve seen them so I can properly cite the source.