Is it obvious that in $\mathbb R^n, \forall \ p,q \ \text{ s.t. } \ p \leq q$
$$ \vert \vert x\vert \vert_{p} \leq \vert \vert x\vert \vert_{q}, \text{where }\vert \vert x\vert \vert_{q} = (\sum_0^n x^q)^{\frac 1 q }$$ If yes, how to prove it ? I was thinking about the classical inequalities like Minkowski, Holder or Cauchy Schwartz but none seems to work here.
Moreover, is this inequality still true for any norm ? Like the norm $L_p $ and $L_q$ ? (I mean by this $ (\int \vert f(x)^q \vert dx ) ^{\frac 1 q } $ )
In other words, is it true for a space of infinite dimension ?
What about the infinite norm ? is it also always bigger than any other one ? (this one I know is false for the case $\mathbb R^n $, but it is true for functions spaces like the space of continuous functions...)
P.S. I'm sorry, I have not yet studied functionnal analysis. I'm asking this for my metric space class where we are proving that some spaces are complete, and I'm wondering when is it possible to use the very usefull inequality of norm I was talking about.
I'm asking this because intuitively, it seems right :
Actually, direction should be reversed: it holds that $$ \|x\|_p\ge \|x\|_q,\quad 1\le p< q\le\infty.\tag{*} $$ What your figure shows us is that $$ B_p\subset B_q,\quad p< q\tag{**} $$ where $B_r$ is a unit ball in $r$-norm. It is easy to see that $\text{(*)}$ is equivalent to $\text{(**)}$.
Now, we prove $\text{(*)}$. By the homogeneity of norms, it is sufficient to show that $$ \|x\|_q\le 1 $$ provided that $\|x\|_p= 1$. By the assumption, we have $$ |x_i|^p\le |x_1|^p+|x_2|^p+\cdots +|x_n|^p=1 $$ for all $i\le n$. Hence $|x_i|\le 1$ and it follows $$ |x_i|^q\le |x_i|^p,\quad i=1,2,\ldots,n $$ for $q<\infty$, and $\|x\|_\infty \le 1$ for $q=\infty$. Summing over $i$ yields $$ \|x\|_q^q =|x_1|^q+|x_2|^q+\cdots +|x_n|^q\le |x_1|^p+|x_2|^p+\cdots +|x_n|^p=1 $$ for $q<\infty$. This proves $\|x\|_q\le 1$ for all $1\le p< q\le \infty$ as desired.
Note: The inequality can be easily extended to $\Bbb R^{\Bbb N}$ equipped with $l^p$-norms, i.e. $l^p(\Bbb N)$. And reverse inequality is true if we are on a finite measure space, i.e. $$ \|f\|_p\le \mu(\Omega)^{1/p-1/q}\|f\|_q,\quad 1\le p<q\le\infty. $$ holds for $f\in L^q(\Omega)\subseteq L^p(\Omega)$. (It a direct consequence of Jensen's inequality.) But in a general measure space, in particular with infinite total measure such as $L^p(\Bbb R)$, none of the relationship $$ \|f\|_p\le C\|f\|_q $$ nor $$ \|f\|_q\le C\|f\|_p $$ is true for any $C>0$. (So there is no inclusion between $L^p(\Bbb R)$-spaces. I'll omit further explanation, but one maybe able to find examples of this.) Thus the fact that $$l^p(\Bbb N)\subseteq l^q(\Bbb N), \quad 1\le p<q\le \infty$$ is a special property, which can be attributed to certain properties of counting measure.