I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ||ab||_1 = \sum_{i}|a_ib_i| \leq ||a||_p||b||_q = (\sum_i |a_i|^p)^{1/p}(\sum_j |b_j|^q)^{1/q}, $$ for $1/p + 1/q = 1$.
1) Proof that $l^p$ is a normed space. I guess I should use this inequality when showing that for arbitrary $x$ and $y$ $$||x+y||_p\leq||x||_p +||y||_p,$$ but I really can't see how to use Hölder's inequality as it involves products rather than sums.
2) Proof that $l^q\subseteq (l^p)^*$, where the star denotes the dual space. I guess I should prove that for an arbitrary $x\in l^q$ we have that $x$ defines a map $x:l^q \rightarrow \mathbb{R}$, but I have no clue how to do this.
$$\begin{align*} \| x+y \|_p^p &= \sum_{i=1}^{n} |x_i+y_i|^p\\ &= \sum_{i=1}^{n} |x_i + y_i||x_i+y_i|^{p-1} \\ &\leqslant \sum_{i=1}^{n} |x_i||x_i+y_i|^{p-1}+\sum_{i=1}^{n} |y_i||x_i+y_i|^{p-1} \tag{1}\end{align*}$$
Then using Hölder inequality on $(p,q)=\left(p,\frac{1}{1-\frac{1}{p}}\right)=\left(p,\frac{p}{p-1}\right)$ (from $\frac{1}{p}+\frac{1}{q}=1$) we obtain $$\begin{align*} \sum_{i=1}^{n} |x_i||x_i+y_i|^{p-1} &\leqslant \left(\sum_{i=1}^{n} |x_i|^p\right)^{\frac{1}{p}} \left(\sum_{i=1}^{n}(|x_i+y_i|^{p-1})^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}}\\ &= \left(\sum_{i=1}^{n} |x_i|^p\right)^{\frac{1}{p}} \left(\sum_{i=1}^{n}|x_i+y_i|^p\right)^{\frac{p-1}{p}}\\ &=\|x\|_p \|x+y\|_p^{p-1}\tag{2}\end{align*}$$
Similarly we find $$\sum_{i=1}^{n} |y_i||x_i+y_i|^{p-1} \leqslant \|y\|_p \|x+y\|_p^{p-1}\tag{3}$$ Putting $(2)$ and $(3)$ in $(1)$ we find $$\| x+y \|_p^p \leqslant \|x\|_p \|x+y\|_p^{p-1}+ \|y\|_p \|x+y\|_p^{p-1}\tag{4}$$ Assuming $x+y \neq 0$ and dividing $(4)$ with $\| x+y \|^{p-1}_p$ we get $$\|x+y\|_p\leqslant \|x\|_p +\|y\|_p$$ If $x+y=0$ then inequality is obvious.