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Math Help - Cauchy-Schwarz Inequality (complex analysis)

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    Cauchy-Schwarz Inequality (complex analysis)

    Nothing makes me stare off into space and wonder why I'm bothering with mathematics at all more than a proof presented with the intent of educating the reader yet the opening statement of which lacks substantive intuitive relevance. So, the proof I want to understand be damned, I come asking for your enlightenment.

    The proof is of the Cauchy-Schwarz inequality for complex numbers (its from a complex analysis text, preface says it's intended for a first graduate course). Let z_j,\,w_j be complex numbers and show that \,\,|\sum_1^n z_j w_j |^2 \leq\,\, \sum_1^n |z_j|^2 \sum_1^n|w_j|^2. The proof begins:

    For any complex number \lambda,\, 0 \leq \sum_1^n|z_j - \lambda \bar{w_j}|^2.

    Would someone kindly explain how I ought to "know", a priori, to introduce an extraneous complex number, the conjugate, essentially this leading inequality and all it involves, to begin the proof of the CS inequality?
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    Quote Originally Posted by cribby View Post
    Nothing makes me stare off into space and wonder why I'm bothering with mathematics at all more than a proof presented with the intent of educating the reader yet the opening statement of which lacks substantive intuitive relevance. So, the proof I want to understand be damned, I come asking for your enlightenment.

    The proof is of the Cauchy-Schwarz inequality for complex numbers (its from a complex analysis text, preface says it's intended for a first graduate course). Let z_j,\,w_j be complex numbers and show that \,\,|\sum_1^n z_j w_j |^2 \leq\,\, \sum_1^n |z_j|^2 \sum_1^n|w_j|^2. The proof begins:

    For any complex number \lambda,\, 0 \leq \sum_1^n|z_j - \lambda \bar{w_j}|^2.

    Would someone kindly explain how I ought to "know", a priori, to introduce an extraneous complex number, the conjugate, essentially this leading inequality and all it involves, to begin the proof of the CS inequality?
    I think the answer is that you're not supposed to "know" that this is the way to prove the result. It wouldn't have names as eminent as Cauchy and Schwarz attached to it if it was a simple enough idea for a graduate student (even a bright one!) to think of.

    The best way to understand a proof like this is to read through it fairly quickly, just to try to get the gist of it. Then go through it again more slowly, to absorb the details. After that, you can sit back and admire the cleverness of it.
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