In this post, we study the concept of character, what it is about in abstract harmonic analysis and how to use it Galois theory.

This blog post is intended to deliver a quick explanation of the algebra of Borel measures on RnRn. It will be broken into pieces. All complex-valued complex Borel measures M(Rn)M(Rn) clearly form a vector space over CC. The main goal of this post is to show that this is a Banach space and also a Banach algebra.

In fact the RnRn case can be generalised into any locally compact abelian group (see any abstract harmonic analysis books), this is because what really matters here is being locally compact and abelian. But at this moment we stick to Euclidean spaces. Note since RnRn is σσ-compact, all Borel measures are regular.

To read this post you need to be familiar with some basic properties of Banach algebra, complex Borel measures, and the most important, Fubini's theorem.

In this post, we study the concept of generalised functions (a.k.a. distributions), and let's see how to evaluate the derivative no matter the function is differentiable or not.

You may have known what it is...

Let us say you are a programmer who has been working in big companies for a decade. How does it feel when you want to help someone who starts studying programming from scratch? You may find it makes no sense that he or she cannot understand that, by copying several lines of code on the book, they has successfully made a programme printing "Hello, world!" on the screen. You know what I am talking about - the curse of knowledge.

The ring

Throughout we consider the polynomial ring R=R[cosx,sinx].R=R[cos⁡x,sin⁡x]. This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.

Left shift operator

Throughout we consider the Hilbert space L2=L2(R)L2=L2(R), the space of all complex-valued functions with real variable such that f∈L2f∈L2 if and only if ∥f∥22=∫∞−∞|f(t)|2dm(t)<∞‖f‖22=∫−∞∞|f(t)|2dm(t)<∞ where mm denotes the ordinary Lebesgue measure (in fact it's legitimate to consider Riemann integral in this context).

For each t≥0t≥0, we assign an bounded linear operator Q(t)Q(t) such that (Q(t)f)(s)=f(s+t).(Q(t)f)(s)=f(s+t). This is indeed bounded since we have ∥Q(t)f∥2=∥f∥2‖Q(t)f‖2=‖f‖2 as the Lebesgue measure is translate-invariant. This is a left translation operator with a single step tt.

Analytic and quasi-analytic vectors

Guided by researches in function theory, operator theorists gave the analogue to quasi-analytic classes. Let AA be an operator in a Banach space XX. AA is not necessarily bounded hence the domain D(A)D(A) is not necessarily to be the whole space. We say x∈Xx∈X is a C∞C∞ vector if x∈⋂n≥1D(An)x∈⋂n≥1D(An). This is quite intuitive if we consider the differential operator. A vector is analytic if the series ∞∑n=0∥Anx∥tnn!∑n=0∞‖Anx‖tnn! has a positive radius of convergence. Finally, we say xx is quasi-analytic for AA provided that ∞∑n=0(1∥Anx∥)1/n=∞∑n=0∞(1‖Anx‖)1/n=∞ or equivalently its nondecreasing majorant. Interestingly, if AA is symmetric, then ∥Anx∥‖Anx‖ is log convex.

Based on the density of quasi-analytic vectors, we have an interesting result.

(Theorem) Let AA be a symmetric operator in a Hilbert space HH. If the set of quasi-analytic vectors spans a dense subset, then AA is essentially self-adjoint.

This theorem can be considered as a corollary to the fundamental theorem of quasi-analytic classes, by applying suitable Banach space techniques in lieu.

We study the concept of quasi-analytic functions, which are quite close to being analytic.

I think it's quite often that, when you are learning mathematics beyond linear algebra, you are stuck at some linear algebra problems, but you haven't learnt that systematically before although you wish you had. In this blog post we will go through some content that is not universally taught but quite often used in further mathematics. But this blog post does not serve as a piece of textbook. If you find some interesting topics, you know what document you should read later, and study it later.

This post is still on progress, neither is it finished nor polished properly. For the coming days there will be new contents, untill this line is deleted. What I'm planning to add at this moment:

• Transpose is not just about changing indices of its components.
• Norm and topology in vector spaces
• Representing groups using matrices