/sci/ - Science & Math

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>>12128651
[math]\left(\sum_{j=1}^n|x_jy_j|\right)^2 \leq \sum_{j=1}^n|x_j|^2 \sum_{j=1}^n|y_j|^2[/math]

So assuming you are dealing with positive real numbers [math]a_j,b_j[/math]
Let [eqn]x_j = \frac{a_j}{\sqrt{b_j}}, \quad \text{ and } \quad y_j = \sqrt{b_j}[/eqn]
Then you easily obtain
[eqn]\left(\sum_{j=1}^n \left|\frac{a_j}{\sqrt{b_j}}\sqrt{b_j}\right|\right)^2 \ \leq \left(\sum_{j=1}^n\left|\frac{a_j}{\sqrt{b_j}}\right|^2\right)\left(\sum_{j=1}^n\sqrt{b_j}^2\right)[/eqn]

From here your equation follows easily.

>>12032464
Can anyone point me to a characterization of epimorphisms in the category of normal topological spaces?

I was hinted to use Urysohn's lemma. but all i can think of is that the forgetfull functor from Top to Set is faithfull and hence reflects epimorphisms so that epimorphisms are precisely surjective.