Motivation: Open Mapping Theorem

Question

One of the Fundamental Theorems in Functional Analysis is the Open Mapping Theorem.

Theorem. Let $X,Y$ be Banach spaces and $T \in L(X,Y)$. Then $T$ is surjective if and only if it is open.

I'd love to see some motivation on this theorem (not a proof!). Why should it be true? How was it discovered?

Perhaps the problem is that one (at least in the beginning of mathematical studies) not often thinks about open maps - but I find it fascinating that people saw a connection between surjectivity and open maps.

Answer 1

One way is to note that the theorem holds for many simple examples. Take for example $X=Y=\ell^1$ with $T$ a multiplication map, multiplying pointwise with some $t\in\ell^\infty$: Then $T$ is open if and only if it is invertible, equivalently, iff $t_k$ is bounded below. Now if $t_k$ is not bounded below, it is a fairly easy exercise to find some $y\in\ell^1$ which is not in the image of $T$: You just need to have $(t_1^{-1}y_1,t_2^{-1}y_2,\ldots)\notin\ell^1$. (Assuming $t_j\ne0$ for all $j$. The case when some $t_j$ is zero, is trivial to handle.)

After you've worked through some examples of this type, the open mapping theorem begins to look rather plausible.

Answer 2

Think of the finite dimensional case. If a linear map is open then it maps the open unit ball to an open ellipsoid centred at 0. It cannot have a lower dimension else the image is "flat" and not open. By expanding the unit ball the ellipsoid increases with it (linearity of $T$) and eventually covers the whole space.

The point of the theorem is that this is still valid in infinite dimensions. The strategy of most proofs is to show that the unit open ball maps to an open set that contains a small ball centered at 0. So by expanding them, the whole space is covered.

Answer 3

I like very much the version in Rudin's book which, in the case of Banach spaces, says that almost open continuous linear operators $T:X\to Y$ are surjective and open, where almost open means that the closure of $T(B_X)$ ($B_X$ being the unit ball of $X$) contains a ball $\alpha B_Y$ for some $\alpha$.

This condition can be spelled out as follows: There is a constant $C>0$ such that for each $\varepsilon>0$ and each $y\in Y$ the equation $y=T(x)$ has an approximate solution with norm control, i.e., there is $x\in X$ with $\|y-T(x)\|\le \varepsilon$ and $\|x\|\le C\|y\|$. The conclusion of the theorem is then that you always have exact solutions with norm control.

(Instead of the for all $\varepsilon$ it is enough to have this for $\varepsilon=\alpha \|y\|$ some fixed $\alpha \in (0,1)$; if you denote $S(y)$ one of the $\varepsilon$-approximate solutions for $y$, you define recursively the sequence $x_{n+1}=x_n +S(y-T(x_n))$ and prove that it converges -- here the completeness of $X$ is needed -- and that the limit is an exact solution -- here continuity or at least closed graph for $T$ is used.)

This viewpoint of the open mapping theorem (I don't know any book really promoting Banach's theorem in that way) encapsulates concrete approximation-and-correction-procedures (as, e.g., in the classical theorem of Weierstraß about poles of meromorphic functions) in an abstract principle.