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7 votes
1 answer
856 views

Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
Saj_Eda's user avatar
  • 395
5 votes
1 answer
805 views

Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?

I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
D.R.'s user avatar
  • 831
2 votes
1 answer
159 views

A compact embedding claim

Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms $$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$ Let $H_2$ be the weighted Sobolev space with the ...
Ali's user avatar
  • 4,135
2 votes
1 answer
135 views

A non-condensing operator with a power condensing

Let $\alpha$ to be the Kuratowski measure of non-compactness, in a Banach space $E$. It's very easy to prove that $\alpha (D_1\times D_2)\leq \alpha (D_1)+\alpha (D_2)$, where $D_1$ and $D_2$ are ...
Motaka's user avatar
  • 291
2 votes
0 answers
170 views

finite dimensionality of a subspace of a Banach space

Let $H$ be the space of measurable functions on $(0,1)$ such that $$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$ Let $C>0$ be a constant. Suppose that $W \...
Ali's user avatar
  • 4,135
2 votes
0 answers
428 views

Weak relative compactness in $L^1_{loc}$.

In my work I stumbled upon a proposition (without proof, alas), which I can't really prove. Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...
TZakrevskiy's user avatar
1 vote
1 answer
123 views

Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$

Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$. To prove that, as $\epsilon \to 0$, the weak solution ...
user avatar
-1 votes
1 answer
102 views

Compactness of a special kind of Integral operators

Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{ & K:{L^2}(0,1) \to {L^2}(0,1) \cr & f: \to (Kf)(x) = \int\limits_0^1 {k(...
Gustave's user avatar
  • 617