Finite Dimensional Hilbert Space

Question

A while ago someone asked this question. I really like what the accepted answer is trying to do. But, I am having trouble figuring out his justification for the first line in the proof: $$\bigcup_{x \in F} B\left(x+ \frac{1}{2} \right) = F+B \left(0,\frac{1}{2} \right)$$

If someone could elaborate, I would really appreciate it.

It is just a notation: ${1,2}+[0,\frac 1 2]:={1+[0,\frac 1 2]}\cup{2+[0,\frac 1 2]}:=[1,1+\frac 1 2]\cup[2,2+\frac 1 2]$ — Marco Disce, Apr 25 '16 at 08:22

score 1 · Accepted Answer · answered Apr 25 '16 at 09:27

The first term contains a typo, it should read $B(x,1/2)$ instead of $B(x + 1/2)$. Moreover, $F + B(0,1/2)$ is the Minkowski sum, which is defined via $$F + B(0,1/2) = \{x + y \colon x \in F, y \in B(0,1/2)\}.$$ Hence, this equality is essentially the definition of the Minkowski sum.

Finite Dimensional Hilbert Space

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