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

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

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

Remark 161.2(Remark).

Recall that an inner product can be defined on an NLS iff the norm satisfies the parallelogram identity. Consider $\mathbf{x}=(1,0,0,\dots )$ and $\mathbf{y}=(0,1,0,0,\dots )$ in ${\ell }_p$.

$$\Vert \mathbf{x}+\mathbf{y}{\Vert }_p^2+\Vert \mathbf{x}-\mathbf{y}{\Vert }_p^2=2\times (1+1{)}^{\frac{2}{p}}=2^{1+\frac{2}{p}}$$

Meanwhile,

$$2(\Vert \mathbf{x}{\Vert }_p^2+\Vert \mathbf{y}{\Vert }_p^2)=2\times (1+1)=4.$$

If the parallelogram law holds, then

$$2^{1+\frac{2}{p}}=4⟹1+\frac{2}{p}=2⟹p=2.$$

Thus, ${\ell }_2$ is the only ${\ell }_p$ space which is an inner product space. An inner product on ${\ell }_2$ can now be obtained from the polarization identity; We prescribe it explicitly and give an alternate proof in Prp 3.

Proposition 161.3(Proposition).

${\ell }_2$ is an inner product space, with the inner product defined by

$$⟨\{x_n\},\{y_n\}⟩=\sum _{n=1}^{\infty }{x}_n{\overline{y}}_n.$$

Proof.

From the Cauchy-Schwarz inequality, we have

$$\begin{array}{rl}{\left|\sum _{i=1}^n{x}_i{\overline{y}}_i\right|}^2& ⩽\sum _{i=1}^n|x_i{|}^2\sum _{i=1}^n|y_i{|}^2\\ & ⩽\sum _{i=1}^{\infty }|x_i{|}^2\sum _{i=1}^{\infty }|y_i{|}^2\end{array}$$

for each $n\in \mathbb{N}$. It follows that ${\sum }_{i=1}^{\infty }{x}_i{\overline{y}}_i$ converges. The remaining properties of Def 87.1 are immediate.□

02d59c

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References

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