LEC ANA2 1

Reviewed metric spaces, vector spaces, normed spaces, and the equivalence of norms in finite dimensional vector spaces.

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Sequence spaces

Definition 161.1(Definition).

Let $1\le p\le \infty$. Define

$$l_p:=\{(a_n{)}_{n=1}^{\infty }\}:a_n\in \mathbb{R}\text{ or }\mathbb{C},\text{ and }\sum _{n=1}^{\infty }|a_n{|}^p<\infty \}.$$

Let $l_{\infty }$ denote $(B(\mathbb{N}),\Vert \cdot {\Vert }_{\infty })$ (this is the space defined here with $X=\mathbb{N}$, which we have shown to be an NLS).

Kumaresan (2005) 1.1.38 proves $l_p$ for $1\le p<\infty$ is a NLS with norm defined by $\Vert (a_n){\Vert }_p:={\left({\sum }_{n=1}^{\infty }|a_n{|}^p\right)}^{1/p}$.

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References

Kumaresan, S. (2005). Topology of Metric Spaces. Alpha Science International Ltd.