I've been wrestling with the concepts of weak and weak-* topologies and how they naturally arise in functional analysis. Despite my efforts, they seem to come out of nowhere for me, and I'm hoping to gain some clarity with your insights. Let’s talk about weak topology first.
According to Folland(Real Analysis: Modern Techniques and Their Applications Page 168)
One often wishes to study the operator $\frac{d}{dx} $, or more complicated operators constructed from it, acting on various spaces of functions. Unfortunately, it is virtually impossible to define norms on most infinite-dimensional functions spaces so that $\frac{d}{dx}$ becomes a bounded operator."
This statement makes sense to me, but I'm struggling to piece everything together. For instance, the concept of a weak derivative kind of feels like we're saying, "Since we're in a framework where we can't distinguish two functions when they coincide almost everywhere (a.e.), let's tweak the notion of derivative to respect this." However, it's not clear to me how a weak derivative becomes bounded when the classical derivative doesn't.
When it comes to different modes of convergence, I can somewhat visualize what's happening. For example, if $f_n \to f$ in $L^1([-1,1])$, it's like saying the average distance between these functions is getting smaller, even though they may differ on a set of measure zero. This kind of intuition works for me for other types of convergence (like in measure, pointwise, almost everywhere pointwise, uniform, almost everywhere uniform, etc.).
But when someone says $f_n \to f$ in the weak topology, I'm lost. I can't intuitively grasp what's actually happening.
Moving on to weak-* convergence, I understand it as a way to naturally embed a Banach space $X$(or make it isomorphic to a subset) into $X^{**}$, its double dual. Sure, it's neat that $X$ lies within $X^{**}$, aligning with our intuitions from finite-dimensional spaces, but I'm questioning if there's more to it. Is there a deeper insight I'm missing?
I've been stuck on this for a while and would greatly appreciate any explanations or insights you could share.
Thank you very much!
P.S:I highly recommend you to use $d/dx$ as an example, especially in solving differential equations since it seems to me a key example.
Best Answer
It seems to me there are two different questions here:
Now, let's try to get an intuition for $1$. Instead of working directly with differential operators, let's start with a minimization problem in calculus of variations:
The setting: a bounded open set $\Omega\subset \mathbb R^n$ with smooth boundary.
Say we want to optimize something like $$I(u):=\int_{\Omega}|\nabla u(x)|^2dx,$$
where $u$ should be a smooth function that is equal to a function $g$ on the boundary of $\Omega$ (let's ignore the second condition for a moment).
First, let us proceed only formally: to find stationary points, we want to have something like $$ 0=\lim_{t\to 0}\frac1t(I(u+t\varphi)-I(u))=2\int_\Omega \nabla u\cdot \nabla \varphi=-2\int \Delta u \varphi=-2\int_{\Omega}u\Delta \varphi,$$ where $\varphi$ is a smooth function with support contained in $\Omega$. Notice that the RHS is defined for a larger class of functions than $C^\infty(\Omega)$: this is very useful, since finding minimizers is easier in a larger space. Hence we already have the intuition that we should try to extend our space from smooth functions to something more general, where the RHs makes sense.
Now let's do something more precise: Since we want to minimize $I(u)$, it makes sense that we want to try and find a topology that makes $I$ into a continuous function. The easiest suggestion is also the right one: let's just use the norm $\|u\|=\|u\|_{L^2}+\|\nabla u\|_{L^2}$ on the spae $C^\infty(\Omega)$. There is, however, one big issue: the space is not complete, which is a nontrivial obstacle if we want to try to find the minimizer of $I$ (and not just take an $\inf$). Let's just take the completion of this space then, call it $H$. It is clear that $\nabla$ admits an extension as a continuous operator to $\nabla: H\to L^2$. How does it behave? Well, exactly as it should based on the intuition above:
$$\int (\partial_{x_i} f) \varphi=-\int f \partial_{x_i}\varphi,$$
where $\varphi$ is the same kind of bump function as above. Some work proves that the two definitions are actually equivalent: that is, the norm extension of $\nabla$ is exactly the weak definition. This is how the weak derivative comes into play in analysis (or at least, in calculus of variations): we are mostly interested in seeing how $\nabla u$ interacts (via pairing) with bump functions, hence the weak derivative definition.
Now that we have the definition of weak derivative, let's try to get an intuition for the weak topology. Well, since we want to minimize $I$ on $X:=\{f\in H: Tr(f)=g\}$ (I haven't defined the trace operator: just think of it as the norm extension of the boundary restriction operator on $C^\infty$$^1$), which by the Poincarè inequality is contained in some ball of $H$, there's an idea that looks rather interesting: if the ball were compact and $X$ closed, then the minimum would be very easy to get. Now $X$ is closed (since $Tr$ is continuous), but the ball is not compact in the norm topology. Hence we need another topology on $H$ which makes balls into compact sets. It cannot be stronger than the norm topology but we would still like it to be related to it, so let's look for a weaker topology that turns balls into compact sets. But wait, if we turn to a weaker topology, how can we be sure that $I$ is continuous? Well, we can't, but we don't really need that. It suffices for $I$ to be semicontinuous; and a function is semicontinuous if it's the supremum of continuous functions. Since $I$ is convex, it's the supremum of a family of linear functionals on $H$, which already nudges us in the right direction: we want a topology that makes the balls compact and that keeps every element of $H'$ continuous. And it's easy to see that that is just the weak topology. (This line of reasoning is the basis of the direct method in calculus of variations).
I've cheated a little bit here, since my space is reflexive so there is no difference between $w$ and $w^*$ topologies. But the intuition remains the same in general, with the caveat that I cannot have all three properties at once:
Weak topology is what we get when we remove the second requirement and weak star is what we get when we weaken the third (not all norm continuous functionals, just the ones that come from the primal space)
I have worked with minimization of functional rather than with PDEs, but there is a clear connection through the Euler-Lagrange equations.
There is a second way to get to these notions, more PDE based, which is as follows:
suppose you want to solve a PDE like $\partial_t u+\partial_x u^2=0$ for some boundary condition. This is, in general, rather ugly (the characteristic lines intersect), but if we tweak it a little bit we get a much nicer one: $$\partial_t u+\partial_x u^2=\varepsilon \Delta u.$$
The solutions $u_\varepsilon$ to this equation are indeed smooth and we would like to take some limit as $\varepsilon\to 0$. But then we incur in the same difficulties as above: we need to extend the space (as the limit is usually not smooth) and we need some weaker notion of convergence to make our proofs easier (first prove the limit exists in some weaker norm, then prove the convergence is actually strong) and the weak topology pops out in the same manner as above.