Let be the set of smooth, compactly supported, real-valued functions on a manifold ,
and be its (algebraic) dual space, the set of all linear functionals
.
It is inside of this dual space that we carve out a subset called distributions, to serve as a suitable
generalization of functions for many purposes in analysis. Why don’t we just take this whole space?
The dual only knows as a vector space,
and so it has lost all knowledge of our underlying manifold and the behavior of its smooth functions!
For an example, lets take realized as :
The set of functions is linearly independent in ,
and so we may extend this to a basis (note here I really mean an honest vector space basis - is uncountable!
This is sometimes called a Hamel basis).
Since is a basis, we can define a linear functional by declaring its values on . Let
be the functional which sends and to and sends all other elements of to .
This functional has forgotten important information about the circle and its functions: for instance, if
is any non-sinusoidal element of the basis, we may approximate arbitrary well by some partial sum of its fourier series
But our linear functional sends to , while sending this arbitrarily-good approximation of to the sum of its fourier
coefficients. Thus, while and its partial fourier series look very different “through the eyes of ”,
even though they are almost indistinguishable from the viewpoint of analysis.
The easiest way to force our definition to recall the nature of the functions in
is to build a natural topology on , and then restrict ourselves to linear functionals
which are continuous.
For example, we can topologize using the norm ,
and then see explicitly that the linear functional constructed above is not continuous:
the partial fourier sums converge to , but does not converge to .
In fact, a linear functional is continuous with respect to this topology on precisely if, for all convergent sequences
in , we have . Doing a little more investigation, this rules out all
functionals which are ‘badly behaved’ in ways similar to the above, while still useful things like delta ‘functions’.
Because_ it will prove useful later on, we review this definition here.
Given a point , the delta distribution is the linear functional such that for all ,
.
These are continuous with respect to the sup norm topology, as if , using smoothness (really, just continuity)
we see that these must also converge pointwise, so , and hence for ny .
If has a distinguished point denoted , we write for .
HOWEVER - we inadvertently went too far in our desire to cut down the size of !
There is a kind of inverse relationship between how many functionals are continuous and how fine of a topology is imposed on the domain.
We chose a rather weak topology, meaning it is rather easy for functions to converge in , and through this inverse rerlationship,
continuity becomes a pretty strong requirement. So strong in fact, that our set of continuous linear functionals is not closed under useful operations like differentiation!
(Recall that the derivative of a linear functional is defined by analogy with integration by parts, so
).
As an explicit example, we see that the derivatives of delta distributions are not continuous.
For concreteness let’s once more restrict ourselves to the circle (the following argument goes through similarly for
, after multiplying each of the sinusoids by a smooth function of compact support),
and consider the sequence of functions .
These functions converge to the constant function with respect to the sup norm, but applying gives
So, even though , we have .
If we wish to have our collection of distributions closed under differentiation, we somehow need to expand our current subset.
And, in light of the inverse relationship between continuity and topology, one way to do this is to strengthen the underlying topology,
and make it harder for sequences of functions to converge!
Looking at that calculation we can pinpoint exactly what went wrong: to compute we actually compute
, and even though was convergent by hypothesis, we had no control over the behavior of .
This suggests a new, stronger topology for , we will say that if and only if we have convergence not only of
in the sup norm, but as well.
With respect to this topology, we have fixed our immediate problem: our explicit example no longer poses a problem as the sequence
simply does not converge in anymore! This strengthening of the topology actually fixes all potential issues
for , which now is a continuous linear functional! Indeed, if in this new topology,
Of course, we can’t stop here to celebrate for too long, as we have not fixed the underlying issue: our new space of continuous
linear functionals is still not closed under differentiation.
Indeed, managing to make continuous is kind of a 2-edged sword, on the one hand now has a derivative (yay!) but
on the other hand, we now need to have a derivative as well (uh-oh).
From our experience above, its straightforward to show that is not continuous: since by definition
all we need to do is choose a sequence of functions which converge in our toplogy but does not
(for an explicit example, integrate our earlier sequence to get ).
So to widen our class of continuous functionals further, we need to again strengthen our topology, and prevent any pathological
sequence like this from being continuous.
Continuing our patchwwork fix, we may suggest requiring that and all converge in
the sup norm for the sequence to count as convergent.
And, of course, this will make continuous, but simply push back the problem once more to its derivative.
Instead, we can attempt to be brave and ‘jump to the end of the line’ and think about what would happen if we applied this patchwork
process infinitely many times.
The resulting topology would require that if , all sequences of derivatives must converge, .
(Here, for 1-dimensional domains is just the number of derivatives, but in general is a multi-index: so for functions on a surface we
are requiring the convergence of things like ).
This has certainly fixed all the problems we were aware of (for instance, is continuous for all now), but
this is real analysis, and its usually the problems that you aren’t yet aware of that mess everything up!
Happily, this time we have truly done it - there are no more unexpected problems out to get us, and the set of continuous
linear functionals with respect to this new strengthened topology really is closed under differentiation! Let’s prove it:
Let be an arbitrary continuous linear functional, and be its derivative. We wish to show that is
continuous, by showing that for any sequence we have .
Given such a sequence , we compute for each the quantity using the definition
of the derivative for linear functionals.
Crucially - our new topology implies that since converged to , the sequence of functions
converges to .
(Note we do not mean merely that in the sup norm, but rather that and all of its derivatives
converge to and all of its derivatives, respectively).
Thus, using the assumed continuity of and the convergence of , we see that .
Putting it all together,
So is continuous as claimed.
This argument makes me think of the space of distributions as being defined by finding a ‘Goldilocks’ topology on .
At first we tried the entire dual space - but it was too big!
Then we tried to cut it down to remember analytical properties of our function space, but the obvious way to cut it down made it too small!
Then, after a sequence of patching the holes we created, we end up with a topology that’s ‘just right’: it remembers pointwise convergence
but is also closed under differentiation.