On the intuition behind the projection theorem.

Question

I have recently proved the projection theorem in a Hilbert space setting. The statements were:

If $M$ is a closed subspace of a Hilbert space $H$ and $x \in H$, then:

1. There is a unique element $\hat{x}$ s.t.

$$\|x-\hat{x}\| = \inf_{y \in M}\|x-y\|$$

2. We have $$\|x-\hat{x}\| = \inf_{y \in M}\|x-y\|$$ if and only if $(x-\hat{x}) \in M^\perp$.

Now I am having some troubles with the intuition behind this theorem, point (1) is intuitive to me but point (2) seems a bit strange even if I have proved it, could someone help me with some intuition or provide a reference that explains the intuition.

Answer 1

Think about this. Suppose you have a point $P$ and a plane in space and that the point does not lie on the plane. There is a unique point $P_0$ of the plane closest to the point. The line segment joining $P$ to $P_0$ is perpendicular to the plane. Do you see why?

It's the same reason a dropped coffee cup falls straight down at the Earth.

Answer 2

Imagine $M$ is the ordinary $x$-axis you learned about in high school. What point in $M$ is closest to $(x,y)$? The closest point is $(x,0)$. And the vector $(x,0)$ (think of it as an arrow from $(0,0)$ out to $(x,0)$) is perpendicular to the vector $(x,y)-(x,0)=(0,y)$ .

Answer 3

$\hskip2in$

I will let you fill in the pieces.

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