All Questions
Tagged with compactness real-analysis
13 questions
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 \...
- 4,145
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 ...
- 4,145
0
votes
0
answers
155
views
Implicit function theorem on curves
I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
11
votes
3
answers
890
views
Structure theorems for compact sets of rationals
Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
- 1,882
5
votes
1
answer
805
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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 ...
- 831
0
votes
1
answer
51
views
A MNC with maximum property but not singular
Let $E$ be a Banach space, $\mathfrak{M}_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the family of all relatively compact sets, and $Ker \mu=\{X\in \mathfrak{M}_E$ ...
- 291
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 ...
user139844
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 ...
- 291
-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 $$\eqalign{
& K:{L^2}(0,1) \to {L^2}(0,1) \cr
& f: \to (Kf)(x) = \int\limits_0^1 {k(...
- 617
7
votes
1
answer
856
views
Compactness of set of indicator functions
Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...
- 395
1
vote
0
answers
71
views
Continuous injection of metric ball into Euclidean ball
This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...
- 5,405
2
votes
0
answers
206
views
Regularity of Dirac measure on Baire sets [closed]
Suppose $X$ is a locally compact Hausdorff space.
Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$,
to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$.
...
- 373
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 $M(v)$ ...
- 173