# 4.13. Sequence Spaces#

We shall assume the field of scalars $$\FF$$ to be either $$\RR$$ or $$\CC$$.

## 4.13.1. The Space of all Sequences#

Recall that a sequence is a map $$\bx : \Nat \to \FF$$ and is written as $$\{ x_n \}$$. The set of all sequences of $$\FF$$ is denoted by $$\FF^{\Nat}$$ or just $$\FF^{\infty}$$ in Cartesian product notation.

Definition 4.150 (Zero sequence)

The zero sequence is defined as:

$\bzero = (0, 0, 0, \dots).$

Definition 4.151 (Vector addition of sequences)

Let $$\bx = \{ x_n \}$$ and $$\by = \{ y_n \}$$ be any two sequences in $$\FF^{\infty}$$.

Their vector addition is defined as:

$\bx + \by \triangleq \{ x_n + y_n \}.$

Definition 4.152 (Scalar multiplication of sequence)

Let $$\bx = \{ x_n \}$$ be any sequence in $$\FF^{\infty}$$ and let $$\alpha \in \FF$$.

The scalar multiplication of $$\alpha$$ with $$\bx$$ is defined as:

$\alpha \bx \triangleq \{ \alpha x_n\}.$

Theorem 4.159

The set of sequences $$\FF^{\infty}$$ is closed under vector addition and scalar multiplication defined above.

This is obvious from definition.

Definition 4.153 (Vector space of all sequences)

The set $$\FF^{\infty}$$ equipped with the vector addition and scalar multiplication defined above is a vector space. It is known as the space of all sequences.

Definition 4.154 (Sequence space)

Any linear subspace of the space of all sequences $$\FF^{\infty}$$ is known as a sequence space.

## 4.13.2. The Space of Absolutely Summable Sequences#

Definition 4.155 (Absolutely summable sequence)

A sequence $$\{x_n\}$$ of $$\FF$$ is called absolute summable if

$\sum_{n=1}^{\infty} |x_n| < \infty.$

If sequences $$\{x_n \}$$ and $$\{ y_n\}$$ are absolutely summable, then their sum $$\{ x_n + y_n \}$$ is absolutely summable with

$\sum_{n=1}^{\infty} |x_n + y_n| \leq \sum_{n=1}^{\infty} |x_n| + \sum_{n=1}^{\infty} |y_n|.$

Proof. Consider the partial sum:

$S_n = \sum_{k=1}^{n} |x_k + y_k| \leq \sum_{k=1}^{n} (|x_k| + |y_k|) = \sum_{k=1}^{n} |x_k| + \sum_{k=1}^{n} |y_k|.$

Taking the limit

$\lim_{n \to \infty} S_n \leq \lim_{n \to \infty}\sum_{k=1}^{n} |x_k| + \lim_{n \to \infty} \sum_{k=1}^{n} |y_k| = \sum_{n=1}^{\infty} |x_n| + \sum_{n=1}^{\infty} |y_n|.$

Thus, the sequence $$\{x_n + y_n\}$$ is absolutely summable.

Theorem 4.161 (Closure under scalar multiplication)

If the sequence $$\{x_n \}$$ is absolutely summable, then for any $$\alpha \in \FF$$, the sequence $$\{ \alpha x_n \}$$ is absolutely summable with:

$\sum_{n=1}^{\infty} | \alpha x_n| = | \alpha| \sum_{n=1}^{\infty} | x_n|.$

Proof. Consider the partial sum:

$S_m = \sum_{n=1}^{m}| \alpha x_n| = \sum_{n=1}^{m} | \alpha | | x_n| = | \alpha | \sum_{n=1}^{m} | x_n|.$

Taking the limit:

$\lim_{m \to \infty} S_m = | \alpha | \lim_{m \to \infty} \sum_{n=1}^{m} | x_n| = | \alpha | \sum_{n=1}^{\infty} | x_n|.$

Hence $$\{ \alpha x_n \}$$ is absolutely summable.

Definition 4.156 ($$\ell^1$$ The space of absolutely summable sequences)

Let $$\ell^1$$ denote the set of all absolutely summable sequences of $$\FF$$. Then $$\ell^1$$ equipped with the vector addition and scalar multiplication defined above is a vector space.

The definition is justified since:

• $$\ell^1$$ is closed under vector addition.

• $$\ell^1$$ is closed under scalar multiplication.

• The zero-sequence $$(0, 0, 0, \dots)$$ is absolutely summable and belongs to $$\ell^1$$.

Definition 4.157 (Norm for the $$\ell^1$$ space)

The standard norm for the $$\ell^1$$ space is defined for any $$\bx \in \ell^1$$ as:

$\| \bx \|_1 = \sum_{n=1}^{\infty} |x_n|.$

The $$\ell^1$$ space equipped with the norm $$\| \cdot \|_1$$ is a normed linear space.

Theorem 4.162

The norm defined for $$\ell^1$$ space in Definition 4.157 is indeed a norm.

Proof. [Positive definiteness] It is clear that the norm of the zero sequence $$\| \bzero \|_1 = 0$$. Now suppose that $$\sum_{n=1}^{\infty} | x_n | = 0$$. The sum of a non-negative sequence is zero only if each term is 0. Thus, $$\{x_n \} = \bzero$$.

[Positive homogeneity] Let $$\bx = \{ x_n \}$$ be absolutely summable. From Theorem 4.161, we have:

$\| \alpha \bx \|_1 = \sum_{n-1}^{\infty} | \alpha x_n | = |\alpha | \sum_{n-1}^{\infty} |x_n| = | \alpha | \| \bx \|_1.$

[Triangle inequality] Let $$\bx = \{ x_n \}$$ and $$\by = \{ y_n \}$$ be absolutely summable. From Theorem 4.160, we have:

$\| \bx + \by \|_1 = \sum_{n=1}^{\infty} | x_n + y_n | \leq \sum_{n=1}^{\infty} |x_n| + \sum_{n=1}^{\infty} |y_n| = \| \bx \|_1 + \| \by \|_1.$

Theorem 4.163

$$\ell^1$$ is complete. In other words, every Cauchy sequence of sequences in $$\ell^1$$ converges to a sequence of $$\ell^1$$. Thus, it is a Banach space.