For $1 \leqslant p<\infty$ and a sequence $x=\left(x_{1}, x_{2}, \ldots\right)$, where $x_{j} \in \mathbb{C}$ for all $j \geqslant 1$, let $\|x\|_{p}=\left(\sum_{j=1}^{\infty}\left|x_{j}\right|^{p}\right)^{1 / p} .$

Let $\ell^{p}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C}\right.$ for all $j \geqslant 1$ and $\left.\|x\|_{p}<\infty\right\}$.

(a) Let $p, q>1$ with $1 / p+1 / q=1, x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{p}$ and $y=\left(y_{1}, y_{2}, \ldots\right) \in \ell^{q}$. Prove Hölder's inequality:

$$\sum_{j=1}^{\infty}\left|x_{j}\left\|y_{j} \mid \leqslant\right\| x\left\|_{p}\right\| y \|_{q}\right.$$

(b) Use Hölder's inequality to prove the triangle inequality (known, in this case, as the Minkowski inequality):

$$\|x+y\|_{p} \leqslant\|x\|_{p}+\|y\|_{p} \quad \text { for every } x, y \in \ell^{p} \quad \text { and every } 1<p<\infty$$

(c) Let $2 \leqslant p<\infty$ and let $K$ be a closed, convex subset of $\ell^{p}$. Let $x \in \ell^{p}$ with $x \notin K$. Prove that there exists $y \in K$ such that

$$\|x-y\|=\inf _{z \in K}\|x-z\| .$$

[You may use without proof the fact that for every $2 \leqslant p<\infty$ and for every $x, y \in \ell^{p}$,

$$\left.\|x+y\|_{p}^{p}+\|x-y\|_{p}^{p} \leqslant 2^{p-1}\left(\|x\|_{p}^{p}+\|y\|_{p}^{p}\right) .\right]$$