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Could someone explain to me, using perhaps a very simple example in @d, what we mean by orthogonal projection from space D to space D'? Thanks

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I am not sure about your notation but a simple example goes as follows. Consider $D=\mathbb R^2$, i.e. the 2-dimensional plane and as $D'$ some line through the origin lying in this plane. Now the projection from $D$ to $D'$ is just a map which assignes a point of $D'$ to each point of $D$ as follows:

Take any point in $D$, drop a perpendicular to $D'$, the point where it hits $D'$ is your image.

Note that any point which is originally in $D'$ stays unchanged.

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To make this example even more concrete, let $D$ be $\mathbb{R}^2=\{(x,y)\}$ and $D'=\{(x,y)\in\mathbb{R}^2|y=0\}$. Now the orthogonal projection of $D$ on $D'$ is the map that sends any pair $(x,y)$ to the pair $(x,0)$. –  Samuel T May 8 '12 at 15:14
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