If $u\in L^\infty$, does the inequality $\int_\mathbb R u(x) |v(x)|^2 dx\le \|u\|_{L^\infty} \int_\mathbb R |v(x)|^2 dx$ hold true?

Question

Let $u, v:\mathbb R\to\mathbb R^3$. Suppose that $u\in L^\infty(\mathbb R, \mathbb R^3)$ and $v\in L^2(\mathbb R, \mathbb R^3)$.

Consider the integral $$\int_\mathbb R u(x) |v(x)|^2 dx.$$

The question is that if I can say that $$\int_\mathbb R u(x) |v(x)|^2 dx\le \|u\|_{L^\infty(\mathbb R, \mathbb R^3)} \int_\mathbb R |v(x)|^2 dx = \|u\|_{L^\infty(\mathbb R, \mathbb R^3)} \|v\|_{L^2(\mathbb R, \mathbb R^3)}.$$

I wrote this in an exercise of my last test, but I was warned that the first inequality is wrong. I can not see why.

Anyone could please help me in understanding this?