The exercice asked me first to prove that $( \sum_i x_i y_i )^2 \leq (\sum_i x_i^2)(\sum_i y_i^2)$ ( Cauchy inequality) and i managed to prove but then it asked me in which case we have the equality ($\left(\sum_{i=1}^n x_iy_{i}\right)^2 = \sum_{i=1}^n (x_i)^2 \sum_{i=1}^n (y_{i})^2$) but i don't Know how ????
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well think backwards in your proof when do you see the equality must occur.Also it would be helpful if you show your proof too – Albus Dumbledore Feb 06 '21 at 13:55
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if there is $\lambda \in\mathbb R$ s.t. x_i=\lambda y_i$. – Bruce Feb 06 '21 at 13:55
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@Bruce but how d I make it to this conclusion ? – it's Rayzig Feb 06 '21 at 15:05
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Equality is attained in the Cauchy-Schwartz inequality $\vert \langle x, y \rangle \vert \leq \vert \vert x \vert \vert \cdot \vert \vert y \vert \vert$ if and only if $x$ and $y$ are linearly dependent and this holds in any space which admits an inner product.
Actually Fritz
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Suppose $x,y\neq 0$. $$|\left<x,y\right>|=\|x\|\|y\|\iff |\left<\frac{x}{\|x\|},\frac{y}{\|y\|}\right>|=1\iff \frac{x}{\|x\|}=\pm \frac{y}{\|y\|}.$$
Bruce
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