metric spaces – A bounded set is contained in an open ball

Given metric space $ (X, d) $ .

Statement : A set $ E subset X $ is bounded iff $ exists $ $ x in E $ and $ r in R^{+} $ such that $ E in B_{r}(x,d) $.

In the above statement, the set $ B_{r}(x,d) $ is an open ball.

Is $ B_{r}(x,d) = {y in X | d(x, y) < r } $ or $ B_{r}(x,d) = { y in E | d(x, y) < r } $ , in context of the above statement ?

bash – Process filenames with spaces listed in text file

I have a list of files listed in a text file named “list.txt”, like this:

one file.jpg 
two file.jpg 
three file.jpg

Now I want to copy all this files to another directory, and I try this:

cp $( cat list.txt ) other_directory

but, as you can see, there are space in the files, and then bash breaks the names in two, and no file is copied.

I’ve tried surrounding the names with quote and double quote, and I tried escape with backslash and
space( ), and also combinations of it, to no avail.

"one file.jpg"
'one file.jpg'
one file.jpg

So, what do I do?

Create files with names lines that starts with "Be" from file city.txt(and delete spaces)

txt file contains cities and i want to grab lines starts with Be from city.txt file and create file

spaces – Clicking on dock icon show the menu bar, but it doesn’t focus on the app

I am now running Big Sur and despite the upgrade I still have this problem.
It depends on the app (it happens with Teams for example), but basically when I am on a space, say 1 for example, and the app is opened on another space, say 3, if I click on the dock icon, all I got is the menu bar (File, Edit, etc…) correctly showing, but the focus is still on space 1.

I have to click on the icon another time to see the shifting to the proper ‘space’ and finally see the user interface.

The same thing is also happening with cmd+tab.

I have tried to do some search with no luck.
The only resource I found is an old super user QA, but it’s related to Mountain Lion and there are some proposal but no accepted answer.

measure theory – Can someone help me with this problem about measurable functions and spaces?

Given (X, M, µ) a measurable space, so that µ(X) = 1. Supposing T:X→X es measurable and µ(T^-1(E)) = µ(E) for every E∈M.
Prove that for every E∈M so that µ(E)>0, there exists a natural n so that µ(E∩T^n(E))>0.

Secondly, prove that for every E∈M, the set containig all x∈X so that there exists a natural n0 so that T^n(x) is not in E for every n ≥ n0, has measure equal to 0.

So far, I’ve reached the conclusion that the problem is equivalent to proving that there exists an n so that T^n(E)∈E or viceversa, since you can easily prove that µ(T^n(E)) = µ(E) for every whole number, but I can’t find a way to prove that either.
Some help would be much appreciated, thank you.

ca.classical analysis and odes – Why do the absolute values of functions in Hardy spaces tend to be non-oscillatory?

I’m trying to find a rigorous formulation of an impression/intuitive notion related to Hardy spaces. It seems to me that functions in Hardy spaces tend to have modulus functions which do not oscillate. This apparently generalizes a property of the exponential function. In particular:
$$exp(ikz) = cos(kz) + i*sin(kz)$$
with $sin(kz)$ the harmonic conjugate of $cos(kz)$, and
$$|exp(ikz)|^2 = cos^2(kz) + sin^2(kz) = 1.$$

For many examples of functions $F$ in Hardy spaces, one has
$$F(x) = u(x) + i v(x)$$
with $u(x)^2 + v(x)^2$ very apparently non-oscillatory. For certain specific examples, I am able to prove something. Bessel functions are a good example, I think.

The spherical Hankel function $h_n(z)$ of the first kind admits
the representation
$$h_n(z) = exp(iz) C_n int_0^infty exp(-zt) P_n(1+it) dt$$ with $C_n$ a constant depending on $n$ and $P_n$ the Legendre function of the first kind of order n. This can can be verified for integers n by direct substitution of the (finite) series expansions for each of these functions. This holds for noninteger orders as well, but proving it is a bit harder. It follows from the equivalent formula
$$h_n(z) = C_n int_1^infty exp(izt) P_n(t) dt$$
that $h_n$ is in a Hardy space of functions analytic on the upper half of the complex plane
(not $H^p$ for any $p>0$, though, since $P_n(t)$ belows up at infinity).

It can be shown by direct expansion of the relevant series that
$$|h_n(x)|^2 = 1+ int_0^infty exp(-xt) d/dt P_n(1+t^2/2) dt.$$
The derivative of $P_n(1+t^2/2)$ is positive on $(0,infty)$, so $|h_n(x)|^2$ is completely monotone on $(0,infty)$. An equivalent formula is
$$|h_n(x)|^2 = z int_0^infty exp(-xt) P_n(1+t^2/2) dt.$$

Obviously, it is too much to ask for |F(z)|^2 to be completely monotone for all F in a Hardy space. But is there a general principal here? It is not enough for F to be
in a Hardy space because there are obvious examples where |F| is highly oscillatory.
For example, take
$$F(z) = int_0^infty exp(izt) phi(t) dt$$
with $phi(t)$ a smooth function which is equal to 1 on (1,100000). The function |F(x)|^2
will oscillate rapidly on $(0,infty)$.

I have the strong impression that this must related to a well-known, standard result, but I don’t know where to look for it.

fa.functional analysis – Function spaces satisfying $mathcal{F}(Mtimes N)simeqmathcal{F}(M)otimesmathcal{F}(N)$

Let $M mapstomathcal{F}(M)$ be a map associating topological vector spaces of some type (that I will call “function spaces”) to geometric spaces $M$ of some type.

For $M$, I’m mostly thinking of manifolds with some additional structure, or locally compact topological spaces. $mathcal F$ may or may not be a functor in some way, though it’s better if it’s a contravariant functor. I’m mostly interested in the case where $mathcal{F}(M)$ is a usual function space such as $L^p(M)$, $W^{k,p}(M)$, $mathrm{Meas}(M)$, like in this question, and this one. I want the function spaces of the form $mathcal{F}(M)$ to have some completed tensor product $otimes$.

Question 1: When does it happen that $mathcal{F}(Mtimes N)simeqmathcal{F}(M)otimesmathcal{F}(N)$ and when does it fail and how badly?

The above tensor property, when $mathcal F$ is a functor, would be better intended to hold naturally, i.e. $mathcal F$ is to be a monoidal functor from spaces with their Cartesian product $times$ to function spaces with $otimes$, but the emphasis is not on the categorical aspect.

ct.category theory – Categorical presentation of direct sums of vector spaces, versus tensor products

One way to think about what the monoidal structure on vector spaces is doing is that it is telling us that vector spaces do not really form a category, or not “just” a category: they form a multicategory whose multimorphisms $V_1, dots V_n to W$ are given by multilinear maps $V_1 times dots times V_n to W$. We care a lot about multilinear maps and not just linear maps in practice so this is a very natural thing to do.

Multicategories are a strict generalization of monoidal categories; if the multihom functor $text{Hom}(V_1, dots V_n ; W)$ happens to be representable as a functor of $W$ for all $V_i$ then the representing object can be written $V_1 otimes dots otimes V_n$ and this should define what is called an “unbiased” monoidal category (an axiomatization where we axiomatize all the $n$-fold tensor products at once rather than just binary ones).

In other words, the tensor product is simply describing an additional structure we care about in practice which is not captured by the category structure alone. Similar situations happen all the time, e.g. rings have additional structure given by multiplication which is not captured by addition alone (and it’s not so bad to think of monoidal abelian categories, say, as generalizations of rings). What the category theory tells us is that the notion of direct sum can be defined using only the notion of linear map but the notion of tensor product requires the notion of multilinear map.

gn.general topology – Relative compactness in topological spaces (reference request)

Motivation and context: For a subset $S$ of a metric space $(M,d)$, the following are two very classical compactness results in Analysis:

  • 1a) The set $S$ is compact if and only if each sequence in $S$ has a subsequence that converges to a point in $S$.

  • 1b) The set $S$ is relatively compact (i.e., has compact closure) in $M$ if and only if each sequence in $S$ has a subsequence that converges to a point in $M$.

Now consider the following analogous claims for a subset $S$ of a topological space $X$:

  • 2a) The set $S$ is compact if and only if each net in $S$ has a subnet that converges to a point in $S$.

  • 2b) The set $S$ is relatively compact in $X$ if and only if each net in $S$ has a subnet that converges to a point in $X$.

Assertion 2a) is also a classical result in point set topology. On the other hand, the implication “$Leftarrow$” in 2b) does not hold, in general.

More precisely, the following holds:

  • (i) If $X$ is not Hausdorff, it may happen that $S$ is compact but not closed, and also has non-compact closure. This shows that 2b) fails, in general.

  • (ii) A bit more interestingly, 2b) can also fail in Hausdorff spaces. Indeed, a counterexample can be constructed if we chose $S$ to be an open half disc with one additional point, in the half-disc topology on the upper half plane; this topology is, for instance, described in Example 78 of Steen and Seebach’s “Counterexamples in Topology (1978)”.
    (It is not stated explicitly there that this space yields a counterexample for 2b), but that’s not difficult to see.)

  • (iii) If $X$ is Hausdorff and the topology on $X$ is induced by a uniform structure (equivalently, if $X$ is completely regular), then 2b) does indeed hold.

Assertion (iii) is not extremely difficult to show, but it is not completely obvious, either. Moreover, (iii) is sometimes quite useful in operator theory. So for the sake of citation, the following question arises:

Question (reference request): Do you know a reference where (iii) is explicitly stated and proved?

Related question: This question is loosely related.

reference request – Holomorphic semigroups on $L^1$ spaces

Let $E$ be a locally compact metric space and $mu$ a non-negative Radon measure on $E$ (we also assume that the support is $E$).

I am concerned with holomorphic semigroups on $L^1(E,mu)$. In particular, I assume the situation where the semigroup is determined by a symmetric Markov process on $E$. So, the semigroup is an extension of a holomorphic (contraction) semigroup on $L^2(E,mu)$.

I know that holomorphic (contraction) semigroups on $L^2(E,mu)$ are extended to holomorphic semigroups on $L^p(E,mu)$ with $1<p<infty$. However, under what conditions would the semigroups be extended to holomorphic semigroups on $L^1(E,mu)$. 

I would appreciate if you could tell me the well-known conditions (even if there are strong restrictions).

I don’t have a clear basis, but I think it is correct in the situation where $0$-order resolvents of symmetric Markov processes are bounded linear operators on $L^infty(E,mu)$.