/sci/ - Science & Math

>>14883839
The distance between [math]Q[/math] the plane containing [math]P[/math] with orthogonal vector [math]n[/math] is norm of the orthogonal projection of [math]P - Q[/math] onto [math]n[/math].
So normalize [math]n[/math] for [math]\hat{n}[/math] with unit norm, then take the norm of the inner product [math]| (P - Q) \cdot \hat{n}|[/math].
>>14884386
>Any reason I can't just use
None.
>>14884495
What's the reverse inclusion? [math]\bigcap \{ S + U : U \text{ is a neighborhood of 0}\} \subseteq \overline{S}[/math]?

>a hint
Use regularity for topological vector spaces.

Full proof below:

If [math]x \in \bigcap \{ S + U : U \text{ is a neighborhood of 0}\}[/math] but [math]x \notin \overline{S}[/math] then there's an open neighborhood [math]x \in U[/math] such that [math]U \cap \overline{S} = \emptyset[/math] (because topological vector spaces are regular).

[math]U' = - (U - x)[/math], is an open neighborhood of the origin, hence there's some [math]s \in S[/math] such that [math]x \in s + U' = s - (U - x) = s - U + x[/math]. We translate both sides by [math]x - s[/math] for [math]-s \in -U[/math], then [math]s \in U[/math], a contradiction.