The Group Algebra of A Finite Group and Maschke's Theorem
The Group Algebra of A Finite Group and Maschke's Theorem
Why Does a Vector Space Have a Basis (Module Theory)
First we recall some backgrounds. Suppose
for
Let
and, for any
Note this also shows that
If
First of all let’s consider the cyclic group
which is actually
which gives the fact that
Following
Let
be a abelian group, and be its torsion subgroup. If is non-trival, then cannot be a free module over .
Next we shall take a look at infinite rings. Let
Suppose we have a basis
Hence
I hope those examples have convinced you that basis is not a universal thing. We are going to prove that every vector space has a basis. More precisely,
Let
be a nontrivial vector space over a field . Let be a set of generators of over and is a subset which is linearly independent, then there exists a basis of such that .
Note we can always find such
Proof. Define
Then
But we already know that
By Zorn’s lemma, let
then if
contradicting the assumption that
The Big Three Pt. 1 - Baire Category Theorem Explained
There are three theorems about Banach spaces that occur frequently in the crux of functional analysis, which are called the ‘big three’:
- The Hahn-Banach Theorem
- The Banach-Steinhaus Theorem
- The Open Mapping Theorem
The incoming series of blog posts is intended to offer a self-read friendly explanation with richer details. Some basic analysis and topology backgrounds are required.
The term ‘category’ is due to Baire, who developed the category theorem afterwards. Let
There are some easy examples of nowhere dense sets. For example, suppose
- A set
is of the first category if is a countable union of nowhere dense sets. - A set
is of the second category if is not of the first category.
In this blog post, we consider two cases: BCT in complete metric space and in locally compact Hausdorff space. These two cases have nontrivial intersection but they are not equal. There are some complete metric spaces that are not locally compact Hausdorff.
There are some classic topological spaces, for example
By a Baire space we mean a topological space
such that the intersection of every countable collection of dense open subsets of is also dense in .
Baire category states that
(BCT 1) Every complete metric space is a Baire space.
(BCT 2) Every locally compact Hausdorff space is a Baire space.
By taking the complement of the definition, we can see that, every Baire space is not of the first category.
Suppose we have a sequence of sets
Let
A subset
of is dense if and only if for all nonempty open subsets of .
We pick a sequence of nonempty open sets
For BCT 1,
Now, if
which shows that
For BCT 1, we cannot follow this since it’s not ensured that
For any
BCT will be used directly in the big three. It can be considered as the origin of them. But there are many other applications in different branches of mathematics. The applications shown below are in the same pattern: if it does not hold, then we have a Baire space of the first category, which is not possible.
is uncountable
Suppose
where
Suppose that
is an entire function, and that in every power series has at least one coefficient is
, then is a polynomial (there exists a such that for all ).
You can find the proof here. We are using the fact that
An infinite dimensional Banach space
has no countable basis
Assume that
It can be easily shown that
Since there is no strong reason to write more posts on this topic, i.e. the three fundamental theorems of linear functional analysis, I think it’s time to make a list of the series. It’s been around half a year.