Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
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Analogue of $\ell^2(X)$ over an arbitrary Banach ring
Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...
- 1,485
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22
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Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
- 651
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80
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Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
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reference request: mercer expansion and kernel underlying Sobolev spaces?
Let us define the periodic Sobolev spaces, for $s > n/2$ by
$$
H_{s}([0, 1]^n) = \{f : [0, 1]^n \to \mathbb{R} :\mbox{for}~j\leq s, f^{(j)} |_{\partial[0, 1]^n} \equiv 0, ~~
\int_{[0, 1]^d} (f^{(s)...
- 420
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55
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Strong sub-differentiability of an equivalent strictly convex norm
First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\...
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Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?
This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
- 1,755
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Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
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60
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asymptotic expansions for $C^{1+\epsilon}$operators
I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators.
More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\...
user522705
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Ultraproduct reflexive
Hello I have the following construction: Let $(E,\|\cdot\|_E)$ be a Banach space, $E_n:=E^n$ and $\|x_n\|_n:=\frac{1}{n}\sum_{k=1}^n \|x_n(k)\|_E$ for all $x_n=(x_n(1),...,x_n(n))\in E^n$ and $n\in\...
- 1
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Building representation of an arbitrary umbral calculus
Consider a set of integrable functions on the interval $(0,1)$.
Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function).
In such system the ...
- 10.1k
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211
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Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
- 101
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88
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Intersection of Sobolev Spaces
Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a "nice" boundary. We have the Sobolev spaces $W^{k,2}(\Omega)$, which are all contained within each other: $W^{m,2}(\Omega)\...
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Existence of a measurable maximizer
Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
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A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
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Dependence of functional integral on the function space
In physics, the following functional integral is considered
\begin{gather}
Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf ))
\end{gather}
It is usually said that the integration is performed ...
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59
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Series representation of functions
Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let
$$
V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\}
$$
...
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Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
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Is a bounded measurable convex function above its interior lower semi-continuous convex envelope?
Let $E$ be a locally convex topological vector space, let $C$ be a convex set which matches the closure of its relative interior $\mathring C=\{ x\in C : \forall y\in C,\exists z\in C,~x\in\mathopen]y,...
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Finding an element of Gelfand triple with a designated time derivative
Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
where $V'$ is the dual of $V$ and the inclusions are ...
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The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
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Rescaling of cosine families
First of all, the best wishes for 2024. Recently, I got aware of cosine operator families (in the framework of evolution equations). It is well-known, that operator semigroups can be rescaled (see for ...
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192
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Reference request: an introduction to nuclear spaces
I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
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Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?
The whole theorem goes as follows:
Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying:
$$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
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Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?
(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...
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Kirszbraun-like extension of periodic functions
Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...
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272
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How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?
Already asked in SE but no response, I think it also reasonably belongs here.
https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions
Basically what the title says, plus ...
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252
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Self-adjoint operator with pure point spectrum
Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true:
A has pure point spectrum (i.e., the ...
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Reference needed for powers of semi-group generators
Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$.
For example, if the ...
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Measurability of the weak completion of an orthogonal representation
Let $G$ be a locally compact group and let $\pi$ be a strongly continuous orthogonal representation of $G$ in a real Hilbert space $H$. Denote by $E$ the real Hausdorff locally convex space obtained ...
- 407
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Constants in the entropy number of the Sobolev space
For a Sobolev space with $W^s(\Omega)$, where $\Omega\subset R^d$ is a compact space with smooth boundary, we know that the entropy number satisfies $e(\delta, W^s(\Omega, 1),\|\cdot\|_{L_\infty})\leq ...
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Generator of contraction semigroup
Assume $A$ is a symmetric operator in a Hilbert space, which generates a contraction semigroup (a priori it is not known, whether this semigroup is self-adjoint). Is A then self-adjoint?
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Chapter 2, Section 5 of Chavel's book “Eigenvalue In Riemann Geometry" is about the zero-point distribution of the derivatives of eigenfunctions
In Chapter 2, Section 5 of Chavel's book, regarding the Neumann eigenvalues of the Laplacian in space forms, how did Chavel determine that $T'_{l,j}$ has ($j-1$) zeros? I have consulted books on the ...
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Amenability of $\textrm{w}_0(A)$ for a $C^*$-algebra $A$
Let $A$ be a $C^*$-algebra with only finite dimensional irreducible representations. As in a previous question, let $\textrm{w}_0(A)$ denote the subspace of $\ell^{\infty}(A)$ consisting of all weakly ...
- 2,605
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Non-degenerate representation of a Banach algebra
Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
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When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
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$L_\infty([0,1], \mathbb{C})$ is it isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{C})$?
By a result of Pełczyński, $L_\infty([0,1], \mathbb{R})$ is isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{R})$. That is the case of real valued functions and sequences.
A natural question then is: ...
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Computationally efficient solution for the measure of central tendency minimizing Lp loss for p > 1
We know that the measure of central tendency that minimizes the Lp loss is $\min_c \sum_{i=1}^n |x_i - c|^p$
For $p=1$ (L1 loss), this is the median. For $p=2$ (L2 loss), this is the mean. Both of ...
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Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
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119
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About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
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Reducing subspaces of unitary operators
Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
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Question about a step in the proof of the min-max principle
I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
- 1,305
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80
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Continuity of linear map on tensor product spaces with different norm properties
I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...
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51
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Reparameterizing a function to be linearly bounded
Trying to find a reparameterization of a function from $f(y, z, \ldots)(x)$ to $f(y(a_1), z(a_2), \ldots)(x)$ so that for all $x \in [r, t]$ we have
$$
|f(y(a_1), z(a_2), \dots)(x) - f(y(b_1), z(b_2), ...
- 101
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137
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Convexity of an equivalent norm
Let $X=l_2$ with usual norm $\|\cdot\|_2$. We define a subspace of $X$ as $D=conv (B_{l_2} \cup B),$ where $B = \{ (x_n) \in l_2 : \sum_{n=1}^\infty \frac{n}{2} x_n^2 \leq 1\}$, conv is the convex ...
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Goldstine theorem in quasi-Banach spaces
A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the ...
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56
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Zero flux along lines
I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...
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Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
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The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
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96
views
Closed formula for Laguerre
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Assume $0<\beta<1$. Is there a closed formula for this sum
$$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$
where $b>0$ and $...
- 521
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A direct proof for non-zero limit points of weighted backward shifts
Fix a sequence $(w_1,w_2,\ldots)$ of positive reals such that the linear operator $T: \ell_2\to \ell_2$ given by
$$
T(x_1,x_2,x_3,....)=(w_2x_2,w_3x_3,\ldots) \text{ for all sequences in } \ell_2
$$
...
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