Proving Holder's inequality for sums

Question

I want to prove the Holder's inequality for sums:

Let $p\ge1$ be a real number. Let $(x_{k})\in l_{p}$ and $(y_{k})\in l_{q}$ . Then, $$\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}y_{k}\vert\le\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)^{\frac{1}{p}}\left(\overset{\infty}{\underset{k=1}{\sum}}\vert y_{k}\vert^{q}\right)^{\frac{1}{q}}$$ with $q\in\mathbb{R}$ such that $\frac{1}{p}+\frac{1}{q}=1$ .

Inspired by a proof I've seen before, I attempted at a solution. I would like you to confirm the ideas and to answer the questions, which correspond to steps of the proof I don't know how to justify.

My attempt: Let $p>1$ be a real number. $Let (x_{k})\in l_{p}$ and $(y_{k})\in l_{q}$ .

If $\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)^{\frac{1}{p}}=0$ or $\left(\overset{\infty}{\underset{k=1}{\sum}}\vert y_{k}\vert^{q}\right)^{\frac{1}{q}}=0$ the inequality is trivially true (Question1: Is it? Why?)

In case both are nonzero, we can define the sequences $(z_{k})$ and $(w_{k})$ with

$$z_{k}=\frac{x_{k}}{\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)^{\frac{1}{p}}}\ \text{ and }\ w_{k}=\frac{y_{k}}{\left(\overset{\infty}{\underset{k=1}{\sum}}\vert y_{k}\vert^{q}\right)^{\frac{1}{q}}}$$

Now, $$\overset{\infty}{\underset{k=1}{\sum}}\vert z_{k}w_{k}\vert\le\overset{\infty}{\underset{k=1}{\sum}}\left(\frac{\vert z_{k}\vert^{p}}{p}+\frac{\vert w_{k}\vert^{q}}{q}\right)$$ (by Young's Inequality).

But $$\overset{\infty}{\underset{k=1}{\sum}}\left(\frac{\vert z_{k}\vert^{p}}{p}+\frac{\vert w_{k}\vert^{q}}{q}\right)=\overset{\infty}{\underset{k=1}{\sum}}\left(\frac{\vert x_{k}\vert^{p}}{p\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)}+\frac{\vert y_{k}\vert^{q}}{q\left(\overset{\infty}{\underset{k=1}{\sum}}\vert y_{k}\vert^{q}\right)}\right)\le1$$ (Question2: Is this last inequality true? Why? If it is, the result follows smoothly...)

So $\overset{\infty}{\underset{k=1}{\sum}}\vert z_{k}w_{k}\vert\le1$ . Multiplying both sides by the (positive) term $\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)^{\frac{1}{p}}\cdot\left(\overset{\infty}{\underset{k=1}{\sum}}\vert y_{k}\vert^{q}\right)^{\frac{1}{q}}$ , it's done. $\square$

Answer 1

For completeness I'll leave here my complete proof, using the suggestions in the comments.

Proof: Let $p>1$ be a real number. $Let (x_{k})\in l_{p}$ and $(y_{k})\in l_{q}$ .

If $\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)^{\frac{1}{p}}=0$ or $\left(\overset{\infty}{\underset{k=1}{\sum}}\vert y_{k}\vert^{q}\right)^{\frac{1}{q}}=0$ the inequality is true: This is equivalent to $\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}=0$ or $\overset{\infty}{\underset{k=1}{\sum}}\vert y_{k}\vert^{q}=0$ , and, if any term of one of these series is nonzero, the series would be nonzero (because all terms are zero or positive), so $\vert x_{k}\vert^{p}=0$ or $\vert y_{k}\vert^{q}=0$ for all $k$ . Hence, $\vert x_{k}\vert^{p}\vert y_{k}\vert^{q}=0$ for all $k$.

In case both are nonzero, we can define the sequences $(z_{k})$ and $(w_{k})$ with

$$z_{k}=\frac{x_{k}}{\left(\overset{\infty}{\underset{l=1}{\sum}}\vert x_{l}\vert^{p}\right)^{\frac{1}{p}}}\ \text{ and }\ w_{k}=\frac{y_{k}}{\left(\overset{\infty}{\underset{l=1}{\sum}}\vert y_{l}\vert^{q}\right)^{\frac{1}{q}}}.$$

Now, $$\overset{\infty}{\underset{k=1}{\sum}}\vert z_{k}w_{k}\vert\le\overset{\infty}{\underset{k=1}{\sum}}\left(\frac{\vert z_{k}\vert^{p}}{p}+\frac{\vert w_{k}\vert^{q}}{q}\right)$$ by Young's Inequality.

But $$\overset{\infty}{\underset{l=1}{\sum}}\vert z_{l}\vert^{p}=\overset{\infty}{\underset{l=1}{\sum}}\frac{\vert x_{l}\vert^p}{\Big\vert\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)^{\frac{1}{p}}\Big\vert^p}=\frac{\overset{\infty}{\underset{l=1}{\sum}}\vert x_{l}\vert^p}{\Big\vert\left(\overset{\infty}{\underset{k=1}{\sum}}\vert x_{k}\vert^{p}\right)^{\frac{1}{p}}\Big\vert^p}=1$$ and similarly $$\overset{\infty}{\underset{l=1}{\sum}}\vert w_{l}\vert^{q}=1.$$

Hence $$\overset{\infty}{\underset{k=1}{\sum}}\left(\frac{\vert z_{k}\vert^{p}}{p}+\frac{\vert w_{k}\vert^{q}}{q}\right)=\frac{1}{p}+\frac{1}{q}=1.$$

So $\overset{\infty}{\underset{k=1}{\sum}}\vert z_{k}w_{k}\vert\le1$ . Multiplying both sides by the (positive) term $\left(\overset{\infty}{\underset{l=1}{\sum}}\vert x_{l}\vert^{p}\right)^{\frac{1}{p}}\cdot\left(\overset{\infty}{\underset{l=1}{\sum}}\vert y_{l}\vert^{q}\right)^{\frac{1}{q}}$ , it's done. $\square$