 Solution

Concavity of $\log$ gives $\log (a/p+b/q)\ge p^{-1}\log a+q^{-1}\log b$ because $1/p+1/q=1$. Exponentiating proves Young inequalitywith 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 $\Vert x{\Vert }_p=({\sum }_j|x_j{|}^p{)}^{1/p}$. Positivity, definiteness and homogeneity are immediate. Summing Young inequality for normalized sequences gives Hölder's inequality $\sum |x_j{y}_j|\le \Vert x{\Vert }_p\Vert y{\Vert }_q$. For a finite truncation $z=x+y$, it yieldsDividing 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 $|x_j^{(m)}-x_j^{(l)}|\le \Vert x^{(m)}-x^{(l)}{\Vert }_p$, and let its limit be $x_j$. The sequence norms are bounded by some $C$. For every finite $N$, passage to coordinate limits gives ${\sum }_{j\le N}|x_j{|}^p\le C^p$, so $x\in {\ell }^p$. Given $\epsilon >0$, choose $M$ with $\Vert x^{(m)}-x^{(l)}{\Vert }_p<\epsilon$ for $m,l\ge M$. Letting $l\to \infty$ in finite truncations and then $N\to \infty$ gives $\Vert x^{(m)}-x{\Vert }_p\le \epsilon$ for $m\ge M$. Thus ${\ell }^p$ is complete, hence a Banach space.