= Solution
Concavity of $\log$ gives $\log(a/p+b/q)\geq p^{-1}\log a+q^{-1}\log b$ because $1/p+1/q=1$. Exponentiating proves <Young inequality>
$$
\boxed{a^{1/p}b^{1/q}\leq a/p+b/q,}
$$
with equality precisely when $a=b$. Equivalently it follows from convexity of the exponential.
Define $\ell^p=\{x=(x_j):\sum_j|x_j|^p<\infty\}$ with $\|x\|_p=(\sum_j|x_j|^p)^{1/p}$. Positivity, definiteness and homogeneity are immediate. Summing <Young inequality> for normalized sequences gives <Hölder's inequality> $\sum|x_jy_j|\leq\|x\|_p\|y\|_q$. For a finite truncation $z=x+y$, it yields
$$
\sum|z_j|^p\leq\sum(|x_j|+|y_j|)|z_j|^{p-1}
\leq(\|x\|_p+\|y\|_p)\left(\sum|z_j|^p\right)^{1/q}.
$$
Dividing and passing to larger truncations proves <Minkowski inequality> and shows $x+y\in\ell^p$, so this is a <normed vector space>.
Let $x^{(m)}$ be a norm-Cauchy sequence. Each coordinate is Cauchy because $|x_j^{(m)}-x_j^{(l)}|\leq\|x^{(m)}-x^{(l)}\|_p$, and let its limit be $x_j$. The sequence <norms> are bounded by some $C$. For every finite $N$, passage to coordinate limits gives $\sum_{j\leq N}|x_j|^p\leq C^p$, so $x\in\ell^p$. Given $\varepsilon>0$, choose $M$ with $\|x^{(m)}-x^{(l)}\|_p<\varepsilon$ for $m,l\geq M$. Letting $l\to\infty$ in finite truncations and then $N\to\infty$ gives $\|x^{(m)}-x\|_p\leq\varepsilon$ for $m\geq M$. \b[Thus $\ell^p$ is complete, hence a <Banach space>.]
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