Sequence Spaces

Why

We can view the set of sequences as vector spaces and give them norms.

Let $\C ^\N $ denote the set of complex-valued sequences. Define $\ell ^{\infty} \subset \C ^{\N }$ to be the set of all bounded sequences. That is,

\[ \ell ^{\infty} = \Set*{x \in \C ^{\N }}{\exists M \in \R \text{ with } \abs{x_i} < M \text{ for all } i}. \]

Then $\ell ^\infty$ with componentwise addition and scalar multiplication is a vector space.

Define $\norm{\cdot }_{\infty}: \ell ^{\infty} \to \R $ by

\[ \norm{x}_{\infty} = \sup_{n \in \N } \abs{x_n}. \]

Then $\norm{\cdot}_{\infty}$ is a norm on $\ell ^\infty$.

Let $\ell ^1 \subset \C ^\N $ denote the set of all absolutely summable sequences. In other words, for $x \in \C ^{\N }$, $x \in \ell ^1$ if

\[ \sum_{n = 1}^{\infty} \abs{x_n} < \infty. \]

Then $\ell ^1$ is a vector space with componentwise addition and scalar multiplication.

Define $\norm{\cdot }_{1}: \ell ^{1} \to \R $ by

\[ \norm{x}_{1} = \sum_{n = 1}^{\infty} \abs{x_n} \]

Then $\norm{\cdot }_{1}$ is a norm on $\ell ^1$.