Expanding a norm over a given space

Question

Let $1 \leq p<q \leq \infty$ (p an q are not related)

Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is $n_o$ such that $x_n=0$ whenever $n\leq n_0\}$

Prove that $\|x\|_q \leq \|x\|_p$ for all $x \in \Phi$ and there is no constant $C$ such that $\|x\|_p \leq C\|x\|_q$ for all $x \in \Phi$

How can I show that $\sup_{0\neq x\in \Psi} \frac{\|x\|_p}{\|x\|_q}=\infty$? this proves both inequations but I cant manage to epand and solve the sup function. What is the system here? is it at all a good direction?

Answer 1

Hint: (I assume $q>p$)

1. You can assume that $||x||_p = 1$. Then each $|x_i| \le 1$. So $|x_i|^q \le |x_i|^p$. Now sum this up.

2. Try the sequence $u_n = \frac{1}{n^{1/p}}$