inequality for a family of vectors

Question

Suppose I have two collections of $n$ vectors $\{u_i\}$ and $\{v_i\}$. To be clear, each $u_i$ and $v_i$ is a vector, not a component of a vector. Is it true that $$\left|\sum_i^n\langle u_i,v_i\rangle\right|^2\le\left(\sum_i^n\langle u_i,u_i\rangle\right)\left(\sum_j^n\langle v_j,v_j\rangle\right)?$$It may be worth mentioning that the case $n=1$ reduces to Cauchy-Schwarz.

Answer 1

Hint: Define vectors $$U = \begin{bmatrix}u_1\\u_2\\\vdots\\u_n\end{bmatrix},$$ and $$V = \begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}.$$ Now apply Cauchy-Schwarz inequality on $U$ and $V$.

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Note that $$\langle U,V \rangle =\sum_{i=1}^{n} \langle u_i, v_i \rangle.$$