Hi, we are still trying to find out what kinds of inverse problems are hard in the sense

that they are ill-posed.

And the mathematical framework was that we consider linear operators between Banach

spaces and we consider inverse problems of type F is A of U.

And we still want to understand when such an inverse problem is ill-posed.

So last time we have talked about linearity and continuity of such operators between Banach

spaces and this leads us to classify operators which might lead to difficult inverse problems.

So just looking at that, the only thing that we have to analyze is properties of this operator

here.

So does A bounded, so if the fact that A is bounded or continuous mean that this inverse

problem here is ill-posed?

And that's not true.

For example, F is U, so just taking the identity mapping, is of course not ill-posed.

Because well, if you have data you can just set your parameter to the data, that's your

unique solution and it's obviously stable with respect to perturbation in data.

So that's no problem at all.

This is not an ill-posed problem.

So this operator, the identity mapping, is a linear and bounded map.

So we have just found the inverse problem, well it doesn't really deserve the name, but

we can call it that.

So an inverse problem of this very naive type with a bounded operator which is not ill-posed.

So boundedness itself is a nice property for such operators to have, but it doesn't say

anything about ill-posedness of the underlying inverse problem.

And the correct mathematical property that will be interesting here is compactness.

And that will be our next definition.

So the goal will be to characterize linear inverse problems as those problems which have

this structure where A is a compact operator.

If that's the case then we have an ill-posed inverse problem.

That's what we want to understand in this lecture.

Definition 2.6, if I've called it correctly.

Let X and Y be Banach spaces, then an operator from X to Y is called compact.

If for any bounded sequence Xn in X there exists a subsequence

Xnk such that the sequence of these objects mapped via A, so Xnk, Axnk, such that this

sequence which is a sequence in Y converges in Y.

Okay, let's make an example here.

Let's just take any infinite dimensional Banach space, let X be an infinite dimensional Banach

space, then the identity mapping from X into itself, so we're just using, well we're taking

the object but we're not touching it in any way, this operator is not compact.

So the identity mapping is not compact.

So this is, let's do this in a concrete example.

For example, let's take X equal to small l2 which is the set of all possible sequences

such that the sum of the sequence elements is finite, so it's the squared sum of the

sequence elements.

So we will have to be very careful here because the space itself is a space of sequences and

now we will consider sequences in this space, so we'll have sequences of sequences, so this

is a minor issue.

I'm trying to differentiate those two by saying that objects in X are indexed as a superscript

and sequences of sequences will have this subscript here.

So why is this not a compact operator?

Is enough to consider any bounded sequence in this space, so a bounded sequence of sequences