The best constants $c$ and $C$ such that $c||x||_p \leq ||x||_q\leq C||x||_p$ Taken from Conway's A course in Functional Analysis Chapter 3 Section 1 Problem 3
Problem statement: For $1\leq p < \infty$ and $x = (x_1,...,x_d)$ in $\mathbb{F}^d$, define $||x||_p = \left[\sum_{j = 1}^d|x_j|^p\right]^{1/p}$; define $||x||_\infty = \sup\{|x|_j: 1\leq j \leq d\}$. Show that all of these norms are equivalent. For $1\leq p, q \leq \infty$, what are the best constants $c$ and $C$ such that $c||x||_p \leq ||x||_q\leq C||x||_p$ for all $x$ in $\mathbb{F}^d$.
I went straight to finding constants $c$ and $C$ such that $c||x||_p \leq ||x||_q\leq C||x||_p$ for all $x$ in $\mathbb{F}^d$ since the equivalence of th enormous would immediately follow from this. However, I'm not even sure where to begin. I thought of trying to let $x_j$ all be the same for a possible candidate for $c$ or $C$,but this doesn't seem too promising.
Any tips would be greatly appreciated!
 A: Even though it is tempting, I do not think that Hölder's inequality will help you in this case. Consider this rather obvious inequality (remember that $\lvert x_j \rvert \leq \lVert x \rVert_\infty$ for all $j$):
$$
\lVert x \rVert_p = \left (\sum_{j = 1}^d \lvert x_j \rvert \right)^{\frac{1}{p}} \leq \left( d \lVert x \rVert_\infty^p \right)^\frac{1}{p} = d^\frac{1}{p} \lVert x \rVert_\infty
$$
Now note that there has to be some $\ell \in \lbrace 1, ..., d \rbrace$ such that $\lVert x \rVert_\infty = \lvert x_\ell \rvert$. So:
$$
\lVert x \rVert_\infty = \lvert x_\ell \rvert = (\lvert x_\ell \rvert^q)^\frac{1}{q} \leq \left( \sum_{j = 1}^d \lvert x_j \rvert^q \right)^\frac{1}{q} = \lVert x \rVert_q
$$
If we combine all of this, we get
$$
c \lVert x \rVert_p \leq \lVert x \rVert_\infty \leq  \lVert x \rVert_q
$$
for $c := \frac{1}{d^\frac{1}{p}}$. If you reapply the above reasoning, you get:
$$
\lVert x \rVert_q \leq C \lVert x \rVert_p
$$
for $C := d^\frac{1}{q}$. This means:
$$
c \lVert x \rVert_p \leq \lVert x \rVert_\infty \leq  \lVert x \rVert_q \leq C \lVert x \rVert_p
$$
for the already defined constants $c, C > 0$. This proves the equivalence of the norms $\lVert \cdot \rVert_r$, $r \in [1, \infty]$.
I have not quite yet figured out whether these are the best constants.
