Sequence space

A sequence space is a considered one in mathematics vector space whose elements are sequences of numbers. Many vector spaces occurring in the functional analysis are sequence spaces or may be represented by such. Examples include, inter alia, the important areas such as all bounded sequences, or all against 0 convergent sequences. The sequence spaces offer a variety of possibilities for the construction of examples and may therefore also be considered as a playground for functional analyst.

Introduction

With the vector space of all sequences in ( = or ) is called. Consequences can be component-wise added and multiplied with real or complex numbers. Are such and such consequences and is, so is

It is clear that with these operations is a vector space. Sequence spaces are subspaces of this vector space, which, in order to secure a Mindestreichhaltigkeit, all consequences are at the n -th digit 1 and 0 everywhere else, contain.

The smallest sequence space is thus the subspace generated by the consequences. This is denoted by and is composed of all the consequences that are different only in finite many points of 0. Therefore He is also called the space of finite sequences, whereby each finite sequence thinks continued by zero to an infinite sequence. So the sequence spaces are subspaces of containing.

The fact that the elements of a sequence space are consequences that we as elements of a vector space simply called points or vectors can lead to misunderstandings. Especially when one considers consequences in such spaces, one has to deal with consequences of consequences.

The following standards or systems of norms or semi-norms are defined on sequence spaces. This gives normed spaces and locally convex spaces.

C0 and c

The most well-known sequence spaces are the space of all convergent to 0 episodes and the space of all convergent sequences. Looking at these spaces the supremum norm, ie, we obtain Banach spaces. The space is a subspace of of codimension 1 Refers to the constant sequence that is at any point is equal to 1, the following applies. Are with the component-wise multiplication and declared Banach algebras, even C *-algebras. Next, one can show that is tight. Both rooms are so separable, because the set of all finite sequences with values ​​from or is countable and dense.

ℓ p

It is the space of bounded sequences with the supremum norm. For his

If, as is obtained by the definition of a metric that makes it a complete topological vector space which is not a normed space. For is

The ℓ p-norm defined (this needs the Minkowski inequality), which makes it a Banach space. The subspace is dense and it follows from the separability for. The space is not separable. Indeed, if, as was the result, which is on each component of equal to 1 and 0 otherwise. Then the uncountably many sequences in pairs the distance 1 from each other and therefore can not be separable.

The rooms are a special case of the more general Lp- spaces, when you look at the counting measure on the space.

Among the spaces is the Hilbert space; by the theorem of Riesz - Fischer is up to isometric isomorphism the only infinite-dimensional separable Hilbert space. All rooms are equipped with componentwise multiplication Banach algebras, is an H * -algebra is a C *-algebra, even a von Neumann algebra.

Duality

It is said that the normalized sequence space has the normalized sequence space as a dual space if the following holds:

Since isometry implies Injektiviät, is particularly an isometric isomorphism.

In this sense, the following dualities are:

• ,
• If and, then.

Locally convex spaces

Purely algebraic one has the isomorphisms and. With this model the sum topology, that is the final topology of all inclusions, define what makes this space to a (LF )-space. is determined by the product topology, that is, by the topology of componentwise convergence to a locally convex space.

The above defined duality normalized sequence spaces can be generalized to locally convex spaces, if point 3 is replaced by the following requirement:

• The mapping is a homeomorphism.

Then and.

Köthe spaces

The following goes back to Gottfried Köthe construction of locally convex spaces episode offers a rich arsenal of examples.

Under a Köthe matrix refers to an infinite matrix with the following properties:

• For all matrix elements and for each there is a with.
• For all indices.

With this data now, the following spaces are defined, and it should be:

.

These spaces are called the Köthe matrix defined by the Köthe spaces (or Köthesche disc cavities ), the standards are called the associated canonical norms. Each of these rooms is with the system of canonical norms a locally convex space, even a Fréchet space.

If you choose the Köthe matrix, the matrix of each component is equal to 1, we get back the above defined normed spaces: . By choosing Köthe matrices, whose matrix elements show a particular growth behavior, one can construct examples of all other classes of area.

For example, the following applies:

For a Köthe matrix following statements are equivalent:

• For each is a Montel space.
• Is a Montel space.
• For every infinite subset and every there is a such that.

For a Köthe matrix following statements are equivalent:

• For each is a Schwartz space.
• For each there is a such that.

For a Köthe matrix following statements are equivalent:

• For each is a nuclear space.
• Is a nuclear space.
• For each there is a such that.

As an application of these statements one can construct examples of Montel spaces which are not Schwartz spaces by choosing a suitable Köthe matrix. Such examples are very important in order to bring some order to the zoo of locally convex spaces.

For the matrix is called the space of rapidly falling consequences. This space plays an important role in the theory of nuclear spaces, because after the set of Komura Komura - this room is a generator of all nuclear facilities.