Functional Analysis

Here, you find my whole video series about Functional analysis in the correct order and I also help you with some text around the videos. If you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:

So a metric is a notion of a distance, which can get any set a structure such that one sees how far or close two points are. This is something, we already can do with numbers but now we want to extend this viewpoint even to abstract spaces.

Knowing what a metric should do is very nice, so let’s apply our knowledge to some examples:

Having a metric, which is a notion of distance, we describe what open or closed sets are:

A nice thing we can in a metric space is to describe a lot of properties with the help of sequences. They work the same as sequences with real numbers, for example. Hence, the definition for a convergent sequence and its limit looks indeed similar. You just have to put in the metric when measuring the distance between a member of a sequence and the limit. With that tool in our tool-set we can characterise a closed set by using sequences:

Another notion that one already knows from the real numbers is that of a Cauchy sequence. In general metric spaces, we can define it in the same way: the distance between two members should get arbitrarily small. This means that one would think that such a sequence has a limit. However, this does not have to be the case. This fact only holds in so-called complete metric spaces.

Now we have learnt so much about metric spaces that we can discuss another important structure in functional analysis: Banach spaces.

You see, a Banach space has a lot more structures than just a metric space. We have the whole underlying linear structure from the vector space and are also able to measure length of single vector. Together with the completeness, we have one of the most important object of study in functional analysis. Let’s look at a typical example:

If you set p = 2 in the last example, you might recognise that is looks like the common euclidean norm in $ \mathbb{R}^n $ or $ \mathbb{C}^n $. From the last one, we already know that we are not only able to measure lengths but also angles! For measuring angles in an abstract vector space, we need to define a new structure: an inner product.

In the last video, you learnt what one of the most important objects of study in functional analysis: Hilbert spaces. To get an idea why they are so interesting, let’s look at some examples.

Having an inner product, one is able to connect this to an induced norm and one gets a very useful inequality:

The Cauchy-Schwarz hints to definition of angles, we might know, e.g., in $ \mathbb{R}^3 $. So let’s generalise this idea in an abstract sense:

As an interlude, let’s go back to the basics. We take metric spaces and look at a map between them. Such a map is called continuous if it satisfies the same rules as a continuous function $ f: \mathbb{R} \rightarrow \mathbb{R} $ which you might already know. In general, continuity means that preimages of open sets are again open sets. However, in metric spaces this can be described equivalently by sequences. Hence one calls this often sequentially continuous:

So we learnt that the inner product $ \langle \cdot, \cdot \rangle $ is a continuous map. This will be used a lot! An with this you have seen a first example of an operator. Most of time, when we say “operator” we mean a linear map between normed spaces and often they are also continuous:

Let’s consider an example to get an idea what bounded linear operators really are:

About bounded operators, we will talk a lot in later videos because they occur in many applications and are, therefore, one of the most interesting and important objects of study in functional analysis. By the way, you might wonder what the name “functional” means: This is just a special linear operator that maps into the number field. Especially for Hilbert spaces, these functionals can be very nicely described by the Riesz representation theorem:

For the moment, I want to go back to the beginning when we are talking about metric spaces. There were a lot of topological notions we defined for metric spaces: open sets, closed sets, boundary points and even continuity. However, there is another very important one: compact sets. They generalise some ideas we have for finite sets even into the infinity we deal a lot in function analysis. Therefore the idea is not easy to grasp at the beginning . However, in metric spaces, we can describe it with the help of sequences:

If we now look at sets in a normed spaces, we get the classical result that in a finite-dimensional space all bounded and closed sets are also compact. However, we already learnt that this is not correct in general and in the next video we see that in an infinite-dimensional space we need more. Indeed, the Arzelà–Ascoli theorem tells us what the missing ingredient for compactness is for a special case:

Let’s now apply the Arzelà–Ascoli theorem in the context of so-called compact operators. The name suggests that these operators generalise the well-known matrices or operators in finite-dimensional spaces just a little bit. Indeed, that is the whole idea as we will discuss it in the next video:

This is now a good point to gather one’s breath and to take a break. Before we continue with the interesting topics around operators, we should go back to basics, in particular to the Banach space $ \ell^{p}(\mathbb{N}) $. You might recall that we have proven the completeness but skipped showing the triangle inequality for the $ \| \cdot \|_p $-norm. We did this because there is some technical work needed to show this in full glory. Now after seeing that there are so many interesting things to find when analysing such Banach spaces, we can repay the debts and dive into the technical proofs. This is what we do in the next to videos:

We already deep in this wide field of functional analysis and have most of the basics behind us. The next big topic, we will tackle is the so-called dual theory which deals with dual spaces of normed spaces. However, before we can start talking about these, we first have to be sure that everyone know what isomorphisms for Banach spaces are:

Now let’s discuss dual spaces of normed spaces. We show that they are always Banach spaces:

Dual spaces sound like fun and they are 🙂 The next example shows how dual spaces can be used to switch between some $ \ell^p(\mathbb{N}) $-spaces.

Let’s look at all the cool stuff we have already discovered in functional analysis: We know how to deal with linear bounded operators and we know that continuous and bounded are equivalent terms for linear operators. This is such a nice result that we can ask what happens in a limit process of operators. Is the limit still bounded? The Banach-Steinhaus theorem answers this question for us:

Let’s immediately go to another important result in functional analysis: The Hahn-Banach theorem. It is connected to linear functionals, so the dual space of a normed space. Indeed, the most important corollary from this theorem is that we always find non-trivial linear functionals that vanish on a given closed subspace.

Talking about important results in functional analysis, another one is the famous open mapping theorem:

The most important implication from the open mapping theorem is the bounded inverse theorem:

In the next videos, I want to start with spectral theory. Here, the so-called spectrum for a bounded operator is a generalisation of the eigenvalues of a matrix:

The next video is about the spectral radius. We also show that for bounded linear operators the spectrum can never be empty.

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