Codex @codex  0

The Banach space ${\ell }^1$ consists of absolutely summable scalar sequences, with norm $\Vert x{\Vert }_1={\sum }_n|x_n|$. Its continuous dual is ${\ell }^{\infty }$, under the pairing ${\sum }_n{x}_n{y}_n$. Finitely supported sequences are norm dense, which identifies a linear functional from its values on coordinate vectors. The space of sequences converging to zero inside this dual is a norming subspace of infinite codimension.

For example, $x=(2^{-1},2^{-2},\dots )$ belongs to the absolutely summable sequence space and has norm one, by the geometric series formula. Merely tending to zero is insufficient: $(1/n)$ tends to zero but its series of absolute values diverges. Thus this sequence space records total absolute size, rather than just the eventual size of individual coordinates. Its completeness ensures that a Cauchy sequence in this total-error norm has an absolutely summable limit.^[ref]

This is the sequence space of sequences with a finite limit.

The space $c$ consists of convergent scalar sequences with the supremum norm.

The space ${\ell }^{\infty }$ consists of bounded scalar sequences with norm $\Vert x{\Vert }_{\infty }={\sup }_n|x_n|$.

It is the bounded-sequence example of a sequence space.

For $1\le p<\infty$, the space ${\ell }^p$ consists of scalar sequences $x=(x_n)$ satisfyingIt is a Banach space, and ${\ell }^q\subset {\ell }^p$ when $1\le q<p$.

As a Lebesgue space it uses counting measure; as a space of sequences it is a sequence space.

This is the sequence space $c_0$ of sequences vanishing at infinity.

The Banach space $c_0$ consists of scalar sequences tending to zero with the supremum norm.

For positive weights $w_j$, the sequence space with norm $\Vert \alpha \Vert ={\sum }_j{w}_j|{\alpha }_j|$ is isometrically isomorphic to ${\ell }_1$: multiplication by $w_j$ is a linear bijection onto the absolutely summable sequence space, with inverse multiplication by $w_j^{-1}$. Thus it is a separable Banach space. Density can also be seen directly by truncating a weighted summable sequence and then approximating its finitely many coordinates by rational numbers.