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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.

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.