Serlo: EN: Sequence spaces

{{#invoke:Mathe für Nicht-Freaks/Seite|oben}} The sequence space is a vector space consisting of infinitely long tuples $(x_1,x_2,x_3,\dots )$. The operations on the sequence space are component-wise addition and scalar multiplication.

Motivation

We have already learned about the coordinate spaces $K^n$ for a field $K$ as an example for vector spaces. Here, each element consists of $n$ distinct, that is, finitely many, entries from $K$. For example, $(1,4,2,-1)$ is an element of ${ℝ}^4$. We can also consider infinite tuples. For example, $(1,4,9,16,\dots )=(n^2{)}_{n\in \mathbb{N}}$ is such an "infinite tuple". A better name for infinite tuples is "sequence". If $K$ is the field of real or complex numbers, these are exactly the already known sequences from calculus.

How do we define the vector space operations on the sequences? On $K^n$ we have defined the operations component-wise. We already know that we can also add and scale sequences component-wise. Therefore, we can also define addition and scalar multiplication on infinite sequences over arbitrary fields. This leads us to the conjecture that the set of all sequences with entries in $K$ should form a vector space. We call it the sequence space over $K$.

We will first define the sequence space precisely and then prove that it is indeed a vector space. Then, in the section Subspaces of the sequence space, we will consider examples of subspaces of the sequence spaces over the real and complex numbers, which are important for advanced calculus.

Notation

Let $K$ be a field.

We always write $(x_i{)}_i$ instead of $(x_i{)}_{i\in \mathbb{N}}$ in this article for sequences with elements from $K$.

Definition of a sequence space

In analogy to the coordinate space, we can also define an addition and a scalar multiplication on $\omega$:

Subspaces of the sequence space

The sequence space has some frequently used subspaces. Most of these subspaces can be defined only over the fields $ℝ$ and $ℂ$. They have many applications in functional analysis, where they are part of an important class of examples. In the field of linear algebra over arbitrary fields, the space of sequences with finite support serves as an example in many places. It is the simplest example of a infinite-dimensional vector space and thus can be used as a good example where statements cease to hold, as the vector space if "too large".

The subspace of sequences with finite support Vorlage:Anker

For example, the notation $c_{00}$ for the space of sequences with finite support can be derived like this: This vector space is a subspace of the space of zero sequences over the fields $ℝ$ or $ℂ$. The latter subspace is usually denoted by $c_0$. The $c$ stands for convergence and the $0$ for the fact that we put only zero sequences of the convergent sequences into the vector space. When talking about convergence, the condition that the sequence eventually becomes $0$ is of course significantly stronger than the condition to converge against $0$. Therefore the space of sequences with finite support $c_{00}$ gets an additional zero into the index.

Subspaces from calculus

In the following, we assume that $K\in \{ℝ,ℂ\}$.

Relationships between the subspaces

We have now learned about some subspaces of the sequence space for $K\in \{ℝ,ℂ\}$. This raises the question of what relations exist between them. Most of the conditions we used to construct the subspaces are conditions from calculus. Fortunately, there are already results in calculus that describe implications between the individual conditions. If we translate these implications into the world of sets and vector spaces, we get the following result:

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