Norm on the space of sequences

Question

Given the sequence spaces $\ell^p$ that are defined as:

$$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$

for $\infty > p ≥ 1$, I'm trying to show that $\ell^p$ becomes a normed space with the norm:

$$||a||_p = \left(\sum_{n=0}^\infty |a_n|^p\right)^{1/p}$$

Now it's not to hard to see why $||a||_p$ is positive, definite and homogenous, but I've been struggling with the triangle inequality. How can this be shown? Could this be an application of the Cauchy-Schwarz-inequality or the Minowski-inequality? Or don't I need any of those to show that the triangle inequality is indeed valid? Thanks in advance!

Answer 1

Let $a = (a_n)_{n\in \Bbb N}$ and $b = (b_n)_{n\in \Bbb N}$ be in $\ell^p$. If $\|a + b\|_p = 0$, the inequality is clear, so suppose $\|a + b\|_p > 0$. Since

$$|a_n + b_n|^p = |a_n + b_n| |a_n + b_n|^{p-1} \le (|a_n| + |b_n|)|a_n + b_n|^{p-1}$$

for all $n\in \Bbb N$, then

$$\|a + b\|_p^p = \sum_{n = 1}^\infty |a_n + b_n|^p \le \sum_{n = 1}^\infty |a_n||a_n + b_n|^{p-1} + \sum_{n = 1}^\infty |b_n||a_n + b_n|^{p-1}.$$

By Hölder's inequality (applied with conjugate exponents $p$ and $p/(p-1)$), we have

$$\sum_{n = 1}^\infty |a_n| |a_n + b_n|^{p-1} \le \|a\|_p \|a + b\|_p^{p-1}$$

and

$$\sum_{n = 1}^\infty |b_n| |a_n + b_n|^{p-1} \le \|b\|_p \|a + b\|_p^{p-1}.$$

Hence

$$\|a + b\|_p^p \le (\|a\|_p + \|b\|_p)\|a + b\|_p^{p-1},$$

or

$$\|a + b\|_p \le \|a\|_p + \|b\|_p.$$

Answer 2

How about this: let $\mu$ be the counting measure on $\mathbb N$. Then $$\int_{\mathbb N} |a| \, d\mu = \sum_{n=0}^\infty |a_n|$$ for any sequence $a = (a_n)_{n \in \mathbb N}$. If $a,b \in L^p(\mathbb N,\mu)$, the Minkowski inequality gives you $$ \|a+b\|_{L^p(\mathbb N,\mu)} \le \|a\|_{L^p(\mathbb N,\mu)} + \|b\|_{L^p(\mathbb N,\mu)}$$ which says that $$ \left( \sum_{n=0}^\infty |a_n + b_n|^p \right)^{1/p} \le \left( \sum_{n=0}^\infty |a_n |^p \right)^{1/p} + \left( \sum_{n=0}^\infty |b_n|^p \right)^{1/p}$$ for any $a,b \in \ell^p$.