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I presume what you want to prove is the following. Let $S$ be a nonempty closed subset of $\mathbb{R}^n$. Then if there is a point $y\in\mathbb{R}^n$ and there are at least two points $p$ and $q$ in …

Uncountably infinite as long as $d\ge2$. We can solve the boundary value problem for countinuous functions on the unit sphere $S$. So we get a linear injection from $C(S)$ to $H(B)$. Now the dimensio …

By Taylor's theorem $$\phi(t+h)=\phi(t)+h\phi'(t)+h^2\phi''(t+u(h,t)h)/2$$ where $0\le u(h,t)\le1$. So $$\phi_h(t)=\phi'(t)+h\phi''(t+u(h,t)h)/2.$$ As $\phi''$ is in $C^\infty_c$ it's pretty clear tha …

Take a dense linear subspace $Z$ of $X$ such that $X/Z$ is infinite-dimension (algebraically). Let $x_1,x_2,\ldots$ be a sequence of elements of $X$ whose images in $X/Z$ are linearly indepdendent. We …

In the most general form, for arbitrary ideals over rings, this is false. In the ring $\mathbb{Z}$ let $I_k$ be generated by $2^k$ and let $J$ be generated by $3$. Then $I_k+J=\mathbb{Z}$ for all $k$ …

At least in the real case the buzzword is "spherical codes". https://mathworld.wolfram.com/SphericalCode.html http://neilsloane.com/packings/ The idea is to find as large as possible a set on an $n$-s …

Every discrete space is a metric space. If we consider a measurable cardinal $\kappa$ as a discrete space, then it has an ultrafilter $\mathcal{F}$ in which the intersection of fewer than $\kappa$ ele …

The answer may be affirmative when $V$ is a reflexive Banach space. Each $T_\epsilon$ is a closed bounded convex set. If the intersection of the $T_\epsilon$ is nonempty then each element of this inte …

One big book on distributions is the first volume of Hormander's The Analysis of Linear Partial Differential Operators. This may not be the easiest book to read, but it is comprehensive and a definiti …