Let's state the Hölder's inequality in the following way:
\begin{equation*} |\sum_{k=1}^n x_k y_k| \leq (\sum_{k=1}^n |x_k|^p)^{1/p} (\sum_{k=1}^n |y_k|^q)^{1/q} \end{equation*}
where $1\leq p < \infty$ and $q$ is such that $1/p + 1/q = 1$.
Note that this just becomes the Cauchy-Schwarz for $p=1/2=q$
Proof is as follows:
Consider $y=\log(x)$; x>0.
Since $y$ is a concave function it follows that $$\theta \log a + (1-\theta) \log b \leq \log (\theta a + (1- \theta )b)$$
Taking exponentials $a^{\theta}b^{1-\theta} \leq \theta a + (1- \theta )b$.Now let $a= \frac{a_i}{A}$ and $b=\frac{b_i}{B}$ where $A= \sum a_i$ and $B= \sum b_i$.
Plug the new expressions for $a$ and $b$ and sum them from $i=1$ to $n$ we end up with \begin{equation*} \sum_{i=1}^{n}a_i^{\theta}b_i^{1-\theta} \leq (\sum_{i=1}^n a_i)^{\theta} (\sum_{i=1}^n b_i)^{(1-\theta)} \end{equation*} Make the substitution $a_i^{\theta} = |x_i|$, $b_i^{1-\theta} = |y_i|$ and $1-\theta = 1/q$, $\theta = 1/p$ you would end up with Hölder's inequality.
Q: I couldn't understand the part in bold. What is $a_i$? , $b_i$? Are they the basis for the vector space? If so Does Hölder's inequality hold only for vector spaces? And how did you sum up two parts of an inequality from $i=1$ to $n$? Can we do that?
What is reason behind setting $a=\frac{a_i}{A}$ and $b= \frac{b_i}{B}$?
Thanks in advance.
$a_i$ and $b_i$ are just a very cunning change of variable nothing more.
As you can see $a_i=|x_i|^{1/\theta}$ $b_i=|y_i|^{1/(1-\theta)}$
Next $a_i>0$ and $b_i>0$ so $a_i^\theta b_i^{1-\theta} >0$ and transforming $$a_i^\theta b_i^{1-\theta} \leq \theta a_i A^{\theta-1} B^{1-\theta}+(1-\theta)b_i A^\theta B^{-\theta}$$ into
$$ \sum_i (a_i^\theta b_i^{1-\theta}) \leq \sum_i (\theta a_i A^{\theta-1} B^{1-\theta}+(1-\theta)b_i A^\theta B^{-\theta})$$
is completely licit.
The inequality is finally obtained noticing that $$\sum_i (\theta a_i A^{\theta-1} B^{1-\theta}+(1-\theta)b_i A^\theta B^{-\theta})=A^\theta B^{1-\theta}$$
Proof is obtained noticing that $|\sum_{k=1}^n x_k y_k| \leq \sum_{k=1}^n |x_k| |y_k|$