Examples of LCTVS - Sequence Spaces

Examples of LCTVS - Sequence Spaces

1. The Space of all Convergent Sequences of Real or Complex Numbers

Definition: The Space of all Convergent Sequences of Real or Complex Numbers is denoted by $c$, and is a normed space (and thus is locally convex) with norm $\displaystyle{\| (x_n) \| := \sup_{n \in \mathbb{N}} |x_n|}$ for each [[$ (x_n) \in c.

If $(x_n) \in c$ then $\{ x_n : n \in \mathbb{N} \}$ is bounded so that $\sup |x_n|$ exists.

2. The Space of all Convergent Sequences of Real or Complex Numbers that are Convergent to $0$.

Definition: The Space of all Convergent Sequences of Real or Complex Numbers that are Convergent to $0$ is denoted by $c_0$, and is a normed space (and thus is locally convex) with norm $\displaystyle{\| (x_n) \| = \sup_{n \in \mathbb{N}} |x_n|}$ for each $(x_n) \in c_0$.

Note that $c_0$ is a subspace of $c$.

3. The Space of all p-Summable Sequences of Real or Complex Numbers

Definition: The Space of all $p$-Summable Sequences of Real or Complex Numbers is denoted by $\displaystyle{\ell^p = \left \{ (x_n) : \sum_{n=1}^{\infty} |x_n|^p < \infty \right \}}$, and is a normed space (and thus is locally convex) with norm $\displaystyle{\| (x_n) \|_p := \left ( \sum_{n=1}^{\infty} x_n^p \right )^{1/p}}$.

4. The Space of all Bounded Sequences

Definition: The Space of all Bounded Sequences of Real or Complex Numbers is denoted by $\displaystyle{\ell^{\infty} = \left \{ (x_n) : (x_n) \: \mathrm{is \: bounded} \right \}}$, and is a normed space (and thus is locally convex) with norm $\displaystyle{\| (x_n) \|_{\infty} = \sup_{n \in \mathbb{N}} |x_n|}$.
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