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I'm trying to understand the proof of the Cauchy-Schwarz inequality: for two elements x and y of an inner product space we have

$$\lvert \langle x,y\rangle\rvert \leq\lVert x \rVert \cdot\lVert y\lVert$$

The proof I am reading goes as follows:

We may assume that $y\neq0$ and $\lVert y\rVert=1$. Indeed, the Cauchy-Schwarz inequality holds when y=0. If $y\neq0$ then $z=\frac{y}{\lVert y\rVert}$ has length 1. So if $\lvert \langle x,z\rangle\rvert\leq \lVert x\rVert$ holds then $$\lvert \langle x,z\rangle\rvert=\frac{\langle x,y\rangle}{\lVert y\rVert}\leq\lVert x\rVert$$ from which $\lvert \langle x,y\rangle\rvert \leq\lVert x\rVert \cdot\lVert y\rVert$ follows.

The confusing part is in bold, why does this inequality hold in general?

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    They stated it as a conditional, so it's likely that the rest of the proof is devoted to explaining it.2012-09-29
  • 0
    @TimDuff yes, it seems to be the case, thanks!2012-09-29

2 Answers 2