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In general, in order that the intersection be defined, we should assume that $(U,\|\cdot\|_U)$ and $(V,\|\cdot\|_V)$ are continuously embedded into an ambient Hausdorff TVS $E$. Then, $(U\cap V,\|\cdo …

Taken by a sort of generalisation frenzy I produced the following; I tried to make it as readable as I could. Recall that the upper and lower Dini derivatives are respectively : $$D^*f(x):=\limsup_{y\ …

This is true, $f$ is actually everywhere differentiable. It is the "Limit under the Sign of Derivative" Theorem; it also holds for sequences of maps between Banach spaces (and you may even allow count …

If $\|u_n\|_{1,p}$ does not diverge to $+\infty$, some subsequence $u_{n_j}$ converges weakly-$W^{1,p}(I)$ to some $g\in W^{1,p}(I)$, and still in $L^\infty(I)$ to $f$. So e.g. $u_{n_j}$ converges …

If $i:X\to Y$ is compact, so is $i^*:Y^*\to X^*$; moreover since here $i$ is dense, $i^*$ is injective.

Say $I:=[a,b]$. We may extend $u$ simply putting it constant for $t\ge b$ and $t\le a$, that is $Eu(t):=u((t\vee a)\wedge b)$. This defines a norm-$1$ linear extensor $E$.

For completeness, let's mention a simpler and more general statement: For $\Omega\subset\mathbb{R}^n$ a bounded open set, $k\in\mathbb{N}$ and $0<\beta<\alpha\le1$ there is a compact embedding $$ C^{ …

Let $(g_k)_{k\ge0}$ be a sequence of smooth functions such that $g_k(x)=1$ if $\text{dist}(x,\partial\Omega)\ge 2^{-k}$, $g_k (x)=0$ if $\text{dist}(x,\partial\Omega)\le 2^{-k-1}$ and $0\le g_k\ …

Consider, for $m\ge2$, the function $\varphi_m(x):=x(1-x)^m$ . So $$\varphi_m(0)=0\qquad \varphi_m(1)=0$$ $$ \varphi_m'(0)=1\qquad \varphi_m'(1)=0\ .$$ It is also easy to see that $\| \varphi_m'\ …

In other words, if $A$ is the infinitesimal generator of $T$, the mild solution of the abstract inhomogeneous Cauchy problem $$\begin{cases}\dot v =A v +f\\v(0)=0\end{cases}$$ needs not to be $W^{1,1 …

I will recycle the answer I've been writing into some words of explanation on Alex Gavrilov's example, which is more simple and elegant. We can focus on the first column $p_n$ of an approximating sequ …

I think in your setting $F$ is not compact, unless $f$ is a constant map. Indeed, if $f$ is not a constant map, it coincides on some non-trivial interval $[a,b]$ with a smooth diffeomorphism $g:\mat …

Actually one can find a larger space of $g$, taking into account the Sobolev inequalities: if $f\in W^{k,p}$ and $g\in W^{k,q}$, then for any order of derivation $0\le i\le k$, one has $D^if\in L^{p_i …

Any maximizer $u\in\{ H^1_0(\Omega): \|\nabla u\|_2=1 \}$ of $\|u\|_p$ is a nonconstant, nonnegative function solving $-\Delta u = \lambda u^{p-1}$, with $\lambda=\lambda_p>0$. So if $u$ maximizes bot …

I think Question 1 has a positive answer. Denote $B=:B_0(1)$ and $A_r:=\{r<\|x\|<1\} $ for $0<r<1$. For functions $f\in W^{1,2}_0(B)$ we have a Poincaré inequality on $A_r$ : $$\int_{A_r} f^2dx\le \ …