$l^p$ spaces are increasing in $p$

functional-analysislimitslp-spacessequences-and-seriessupremum-and-infimum

Problem

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $x\in\mathbb{K}^\mathbb{N}$ be any sequence in $\mathbb{K}$. Let the $p$-norm be defined by $$\lVert x\rVert_p:=\left(\sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^p\right)^\frac{1}{p}$$ for $1\leq p\lneq\infty$ and $$\lVert x\rVert_\infty:=\lim_{p\to\infty}\lVert x\rVert_p=\sup_{n\in\mathbb{N}}\left\lvert x_n\right\rvert$$ for $p=\infty$. The $l^p$-space is defined by $$l^p:=l^p(\mathbb{K}):=\left\{x\in\mathbb{K}^\mathbb{N}\ \mid\ \lVert x\rVert_p\lneq\infty\right\}$$

Let $1\leq p \lneq q\leq\infty$. Prove $l^p\subsetneq l^q$.

My Attempt

  • $1\leq p\lneq q\lneq \infty$:

\begin{align*}
&x\in l^p\ \Leftrightarrow\ \lVert x\rVert_p\lneq\infty\\
\Leftrightarrow\ &\left(\sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^p\right)^\frac{1}{p}\lneq\infty\\
\Leftrightarrow\ &\sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^p\lneq\infty\\
\Leftrightarrow\ &\lim_{n\to\infty}\left\lvert x_n\right\rvert^p=0\ \Rightarrow\ \lim_{n\to\infty}\left\lvert x_n\right\rvert=0\\
\Rightarrow\ &\exists N\in\mathbb{N}:\forall n\geq N: \left\lvert x_n\right\rvert\leq 1&\left(p\lneq q,\ 0\leq\left\lvert x_n\right\rvert\leq1\right)\\
\Rightarrow\ &\exists N\in\mathbb{N}:\forall n\geq N: \left\lvert x_n\right\rvert^q\leq\left\lvert x_n\right\rvert^p &\left(\sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^p\lneq\infty\right)\\
\Rightarrow\ &\sum_{n\geq N} \left\lvert x_n\right\rvert^q\leq\sum_{n\geq N}\left\lvert x_n\right\rvert^p\lneq\infty &\left(\sum_{n<N}\left\lvert x_n\right\rvert^q \lneq\infty\right)\\
\Rightarrow\ &\sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^q\lneq\infty\\
\Leftrightarrow\ &\left(\sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^q\right)^\frac{1}{q}\lneq\infty\\
\Leftrightarrow\ &\lVert x\rVert_p\lneq\infty\ \Leftrightarrow\ x\in l^q
\end{align*}

  • $1\leq p\lneq q=\infty$:
    \begin{align*}
    &x\in l^p\ \Leftrightarrow\ \lVert x\rVert_p\lneq\infty\ \Leftrightarrow\ \left(\sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^p\right)^\frac{1}{p}\lneq\infty\\
    &\Leftrightarrow\ \sum_{n\in\mathbb{N}}\left\lvert x_n\right\rvert^p\lneq\infty\ \Leftrightarrow\ \lim_{n\to\infty}\left\lvert x_n\right\rvert^p=0\ \Rightarrow\ \lim_{n\to\infty}\left\lvert x_n\right\rvert=0\\
    &\Rightarrow\ \sup_{n\in\mathbb{N}}\left\lvert x_n\right\rvert\lneq\infty \Leftrightarrow x\in l^\infty.
    \end{align*}
    If $x=(1,1,1,\dots)$, then $\sup_{n\in\mathbb{N}}\left\lvert x_n\right\rvert=1\lneq\infty$, but $\lVert x\rVert_p=\infty$. We have $x\in l^\infty$ while $x\notin l^p$.

Questions

  1. Is everything up to this point correct?
  2. What is an easy example for $x\in l^q$ with $x\notin l^p$ for the case $1\leq p\lneq q\lneq \infty$?

Best Answer

My favourite proof of $\|x\|_p\le \|x\|_q$ for $0<q<p\le \infty$ is the following. By homogeneity it suffices to consider the case $\|x\|_q=1.$ Then $|x_n|\le 1$ and $$\|x\|_p^p =\sum |x_n|^p\le \sum |x_n|^q=1$$ Hence $\|x\|_p\le 1=\|x\|_q.$

My favourite easy example for $\displaystyle x\in \left (\bigcap_{q>p}\ell^q\right )\setminus \ell^p$ is $$x_n= n^{-1/p}$$

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