Inequality between $\ell^p$-norms

Question

Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$). Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into another $\ell^q$?

Answer 1

If $1 \leq p \leq q \lt \infty$ then $\|x\|_{q} \leq \|x\|_{p}$ and clearly $\|x\|_p \geq \|x\|_\infty$. In particular, $\ell^p$ embeds continuously into $\ell^q$ whenever $p \leq q$.

To see this, note that both sides of the inequality $\|x\|_{q} \leq \|x\|_{p}$ are homogeneous in $x$ (multiplying $x$ with a positive real number multiplies both sides with the same positive factor), so we may take without loss of generality an $x$ with $\|x\|_{p} = 1$. Then $\|x\|_{q}^{q} = \sum_{j = 1}^{\infty} |x_{j}|^{q} \leq \sum_{j = 1}^{\infty} |x_{j}|^{p} = 1$, and this is because for $t \leq 1$ and $p \leq q$ we have $t^{q} \leq t^{p}$.

This means that $p \mapsto \|x\|_p$ is decreasing. In terms of spaces, we have the inclusions $\ell^1 \subset \ell^p \subset \ell^q \subset \ell^{\infty}$ whenever $1 \lt p \lt q \lt \infty$ and it is not hard to show that the inclusions are all strict.

Note that this is opposite to the case of finite measure spaces $(\Omega,\mu)$, where the inclusions go the other way around: $L^1(\Omega,\mu) \supset L^p(\Omega,\mu) \supset L^{q}(\Omega,\mu) \supset L^{\infty}(\Omega,\mu)$.

See also the Wikipedia page on $L^p$-spaces.