# Tagged Questions

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### Closure of $C_0^{\infty}$ in $W^{k,p}(\Omega)$

Why is it that in the definition of $W_0^{k,p}(\Omega)$ for $\Omega$ with boundary smooth enough, we only have $D^{\alpha}u$ for all $0\leq|\alpha|\leq k-1$ vanishing at the boundary and not ...
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### What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
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### Prove that $C[a,b]$ with inner product $\langle f,g\rangle:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space.

Prove that $C[a,b]$ with inner product $<f,g>:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space. Now the norm induced by the inner product is \begin{align} ...
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### Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$

Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$. Is it sufficient to take $x=(1,0)$, $y=(0,1)$ in $\mathbb{C}^2$ and just showing that \begin{align} ...
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### when does bijective map exist for any pair of rational function?

Let me ask kind of different questions than former ones. Given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}\text{, and }\frac{P_3(y_1,y_2,\dots,y_n)}{P_4(y_1,y_2,\dots,y_n)}$$ where $P_i$ ...
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### Prove that Hölder condition in $\Bbb R^n$ implies continuity

$f:I\subset \Bbb R^n \rightarrow \Bbb R^m$ is said to be Hölder continuous if $\exists$ $\alpha>0$ and $M>0$ such that $\|{f(x)-f(y)}\| \leq M\|x-y\|^\alpha$, $\forall x,y \in I$, ...
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### Qualitative properties of solutions to a ordinary differential equation.

I have this problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\u(0)=u(+\infty)=0\end{cases}$$ we have that $u$ is continues, $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and ...
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### Show that R is an equivalence relation on X for x, y in X iff f(x) = f(y)

$f:X→Y$ $x,y ∈ X,xRy$ iff $f(x) = f(y)$ Show that R is an equivalence relation on X. Also when $X = Y = \mathbb{R}$ and $f: \mathbb{R} \to \mathbb{R}$ with $x \mapsto x^2$ for all $x∈R$ find the ...
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### A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$ I am trying to prove the triangle inequality of this norm. So far I have that: \begin{align} ...
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### Does compactness preserve continuity? [closed]

Are all compact set continuous in a Banach space? I am clear that continuity preserves compactness, but not clear about the reverse, i.e. does compactness preserve continuity? If not, can someone ...
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### Find a discontinuous linear map on $c_0$

I want to find a discontinuous linear map $\phi: c_0 \to \mathbb{C}$. where $c_0$ has sup norm obviously, $\|.\|_\infty$ I can't think of any example. please suggest me one. I ll try checking it ...
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### If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
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### For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
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### Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...
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### Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
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### Proving the existence of limit of an integral

Let $g:\mathbb{R}^d\to\mathbb{R}$ a smooth function and $B:\mathbb{R}^d\to\mathbb{R}^d$ a Lipschitz continuous vector field. I have to study the limit of the following integral ...
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### Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
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### Sequence of orthogonal vectors in a Hilbert space

Let $\{x_n\}_{n\in\mathbb{N}}$ be a sequance of pairwise orthogonal vectors in a Hilbert space $H$. Show that the following are equavalent: (a) $\sum_{n=0}^\infty x_n$ converges in the norm topology ...
Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
Compute the eigenvalues and eigenfunctions of $$\varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x)$$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...