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functional analysis - $(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right?We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. To that end we consider the transformation $T: (x,y) \mapsto (x + \alpha, y + 2x + \alpha)$ on the $2$-dimensional torus $\mathbb{T}^2$ endowed with the Lebesgue measure $\lambda \times \lambda$.a) Show that the action of $T$ on the torus is ergodic, i.e., if a measurable set $A \subset \mathbb{T}^2$ is invariant under $T$, then $(\lambda \times \la...Read more

functional analysis - Characterization of dual-* topology in terms of subbasis

Preamble: let $(X, \tau)$ be a topological vector space over $\mathbb K$, and $X'$ its (topological) dual space. Then, as I understand, the family of sets given by$$\mathcal B = \{ f^{-1} (V) \mid f \in X' , \, V \text{ open in } \mathbb K \}$$is a subbasis for the weak topology induced by $X'$ on $X$. This topology, of course, has the property of being the coarsest topology on $X$ for which every $f \in X'$ is $\tau$-continuous.Question: Is it possible to make an analogous characterization of the weak-* topology induced by...Read more

functional analysis - Proof idea - closed subspace of compact metric space is compact

I came up with the following proof and I am afraid it is too simple and that I missed something...Question: If $X$ is a compact metric space and $M \subset X$ is closed, show $M$ is compact.Proof idea: We know that $X$ compact gives that $X$ is closed and bounded. To show that $M$ is compact we must just show that $M$ is bounded since it is given that $M$ is closed. But since $M \subset X$ and $X$ is bounded doesn't that immediately give that $M$ is bounded?...Read more

functional analysis - Closed subset of $W^{1,2}([a, b], \mathbb{R})$

In the context of weak solutions of boundary value problems, I want to show that the set $$\{u \in W^{1, 2}([a, b], \mathbb{R}) \; : \; u(a) = 0 = u(b) \}$$ is closed in $W^{1,2}([a, b], \mathbb{R})$. Is there any elementary way of proving it? I might know one way by using the, due Sobolev's embedding theorem, well-defined and continuous dirac delta $\delta_a$ and $\delta_b$ on $W^{1,2}([a, b], \mathbb{R})$. However, I want to refrain from distribution theory. Hints are appreciated....Read more

functional analysis - Polar Body contained in $B_n^2$

Let $K$ is centrally symmetric convex body containing the $\ell_2$ unit sphere $B_n^2$. The polar body of $K$ is define as$$K^{\circ} = \left\lbrace y\in \mathbb{R}^{n} \mid \left< x,y \right> \le 1, \forall x\in K \right\rbrace$$I'm trying to figure out the following question: Let $C$ be the contact points of $K$ with $B_n^2$ (and assume it is not empty). Show that $C \subseteq K^\circ$.I know that $\left(K^{\circ}\right)^{\circ} = K$ and that $\left(B_n^2\right)^{\circ} = B_n^2$. I also know that for $y\in C$ we have $\left\Vert y \r...Read more

functional analysis - Hypothesis in the Radon-Nikodym theorem.

I have a question concerning a proof of Radon-Nikodym theorem here. Why is the hypothesis "$\nu$ is finite" necessary? The author uses it to have the measure $\sigma=\mu+\nu$ finite and then, from $|Tu|\leq \Vert u\Vert_{L^2(X,\mathcal{F},\sigma)}\sqrt{\sigma(X)}$, conclude that $T$ is bounded. By I think that it is also true that $|Tu|\leq\Vert u\Vert_{L^2(X,\mathcal{F},\sigma)} \sqrt{\mu(X)}$ (applying first Hölder's inequality and then the fact that $\mu\leq\sigma$), so $T$ is bounded in any case .So for me, $\mu$ has to be finite, but $\nu$...Read more

functional analysis - Reference request: Choquet theory

Recently I realized that many integral representation theorems (such as Herglotz' theorem, Bernstein's theorem, Riesz representation theorem, etc) may be systematically understood under the Choquet theory.I have never been explicitly exposed to this subject, however, thus I would like to have some good introductory material on it. Any reference that leads to Choquet theorem is fine, but it will be much nicer if it contains some criteria for uniqueness of representation (if any such thing exists) as well as application to some well-known theorem...Read more

functional analysis - Commutative Banach algebra and its maximal ideal space

Let $A:=C^{(n)}([0,1])$ be the set consisting of the n-times continuously differentiable complex-valued functions. Consider $A$ with the norm$$\|f\|:=\max\limits_{0 \leq t \leq 1} \sum_{k=0}^{n}{\dfrac{|f^{(k)}(t)|}{k!}}.$$ I want to show that $A$ is a commutative Banach algebra and find its maximal ideal space.I know that the following holds:for all $f,g\in A$: $(fg)(x)=f(x)g(x)=g(x)f(x)=(gf)(x)$;for all $f,g,h\in A$: $((fg)h)(x)=(fg)(x)h(x)=g(x)f(x)h(x)=f(x)(gh)(x)=(f(gh))(x)$;for all $f,g,h\in A$: $(f(g+h))(x)=f(x)(g+h)(x)=f(x)(g(x)+h(x...Read more

functional analysis - Injective Connecting Maps in Direct Limits of TVS

I am reading Jarchow's Locally Convex Spaces, and I've come across a few questions about direct limits of TVS I have not been able to answer (these questions are based on section 4.5 of Jarchow's book). I start by setting up some notation: let $J$ be a directed set, and for every $j \in J$, let $E_j$ be a TVS. Whenever $j \leq k$, let $S_{kj}:E_j \to E_k$ be a continuous linear map, and suppose that $S_{ki} = S_{kj} \circ S_{ji}$ whenever $i \leq j \leq k$, and suppose $S_{jj}$ is the identity map $E_j \to E_j$. For $i \in J$, let $I_i: E_i \to...Read more

geometry - Can we measure total "closeness" of two geometric objects with analysis?

Say I have one sphere and one plane $$x^2+y^2+z^2 = 1$$$$z = 2$$We can easily calculate closest distance between these. It is easy exercise. Maybe first exercise in differential geometry or something like that. But is there some sense in which we can calculate "total closeness" of the objects? Like integrating over the objects calculating all distances and boil it down to one real number somehow....Read more

functional analysis - Mathematically inclined books on Signal Processing Theory

First off, i know this may seem off topic but i could not find help in signal processing communities so i was hoping there would be people here who both love mathematics and have interest in signal processing. I'm an electronics engineering student with high inclination to analysis and pure mathematics ( abstract algebra/linear algebra ... ). I was just wondering if there was any book ( or any resource ) that treats signal and systems and signal processing with a lot of mathematics rigour ( actually doing proper complex analysis, us...Read more

functional analysis - Multiplication of a distribution by a continuous function

In my work, I am encountering the issue of having to multiply a continuous function (not necessarily differentiable) by a distribution.It seems to me that if $f(x)$ is a continuous function on $\mathbb{R}$ and if $d(x)$ is a distribution on $\mathbb{R}$, then the product $f(x) d(x)$ makes sense and can be interpreted as a distribution. If someone knows an example of a distribution, which cannot be multiplied by a continuous function, I would like to see it. If there is no such example, then my question is this: is there a reference discussing t...Read more

probability theory - Which functional space does feedforward neural network approximate?

I'm pretty sure this question has already studied by at least one paper, but I can't figure out where. The question is the following :Given $l$ layers of $n_l \in L$ neurons, we can build a set of functions $\mathcal{N}(l, \{n_i | 1 \leq i \leq l\}\}$, which are the functions given by any neural feedforward neural network with this amount of layers and neuron (n_1 and n_l are respectively the input layers and output layers).Those functions are continuous $\mathcal{N}(l, n_i) \subset \mathcal{C}(R^n_1, R^n_l)$, and as the number of layers, or th...Read more

functional analysis - Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$:Consider this hypercube $O = H_{R}(x) = \prod_{i=1}^{n}(\alpha_{i},\beta_{i}) = (\alpha_{1},\beta_{1}) \times \prod_{i=2}^{n}(\alpha_{i}, \beta_{i})$ and the following partition of it:We divide $(\alpha_{1}, \beta_{1})$ into intervals of length $(\beta_{1}-\alpha_{1})2^{-m}$ where $m \in \mathbb{N}$. Each of these intervals is then subdivided into two parts of equal length $(\beta_{1}-\alpha_{1})2^{-(m+1)}$...Read more