$\imath ^{p}$ is a normed space

Question

In my notes a given example is as suggested:

Example:

On the space of sequence $\imath$ we define , for $1\leq p< \infty \left \| x \right \|_{p}:=\left ( \sum_{i=1}^{\infty}\left | x_{i} \right |^{p} \right )^{\frac{1}{p}}$ and $\imath ^{p}:=\left \{ x \in \imath : \left \| x \right \|_{p}<\infty \right \}$ is a normed space.

Clearly, $\imath$ is a vector space.

Recall that for any vector space V and a norm $**\left \| \cdot \right \|, \left ( V,\left \| \cdot \right \| \right )**$ is a normed space.

How should I write the normed space in the bold notation?

Is $\left ( l,\left \| x \right \|_{p} \right )$ correct? What about writing $\imath ^{p}$ is a normed space in notation $\left ( V,\left \| \cdot \right \| \right )$? Secondly, how do I show that $\imath ^{p}$ is a normed space?

Thanks in advance.

Answer 1

Notation: There's rarely a false but I like the one of the book you quoted.

To show $l^p$ is a normed space you have to show the axioms of a norm. See here https://en.wikipedia.org/wiki/Norm_(mathematics)

The difficult one is the triangle inequality and you need Minkowski there; if more info needed answer^^

Btw these spaces are called $l^P$-spaces. https://en.wikipedia.org/wiki/Sequence_space