Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,437 questions with no upvoted or accepted answers
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Ordering of completely bounded maps
Let A be a C*-algebra, let H be a Hilbert space, and let $T:A\rightarrow B(H)$ be a completely bounded (cb) map (that is, the dilations to maps $M_n(A)\rightarrow M_n(B(H))$ are uniformly bounded). ...
- 18.7k
6
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2k
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Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
6
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639
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Hilbert subspaces of indefinite inner product spaces
Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...
- 4,060
6
votes
1
answer
897
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Flat norm metrizes the weak* topology
I've come across the following statement in literature (without proof or reference) about the flat norm of currents
$$
F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \...
5
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0
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77
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What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?
Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$.
The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
5
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0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
5
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180
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A variant of Schwarz's theorem on generators of smooth $G$-invariant functions
Let $V$ be a finite dimensional euclidean space and let $G\subset O(V)$ be a finite (or compact) group, let $\mathbb{R}[V]$ be the algebra of polynomial functions on $V$. If $E\subset \mathcal{C}^\...
- 4,123
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94
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When a compact subset of a TVS can be continuously projected on a closed linear subspace?
Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact.
(Q):
When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
- 60.5k
5
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899
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Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?
Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases}
a_k,b_k\in \mathbb{R}\ \forall k=1,\...
- 181
5
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213
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Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
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answers
360
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Injectivity of div–curl operator
$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system
\begin{align*}
Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\
u &= 0 \...
- 419
5
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answers
159
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Log Sobolev inequality uniform in parameters
Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
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5
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194
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When does the Fourier transform of a measure decay?
Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...
- 1,101
5
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417
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All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 1,075
5
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0
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137
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A list of properties of $(\bigoplus \ell^1_n)_{\ell^p}$, $1<p<\infty$
The Banach space $E=(\bigoplus_{n=1}^{\infty}\ell^1_n)_{\ell^p}$ for $1<p<\infty$ shows up in various places in the literature to construct counterexamples. The purpose of this post is to ...
- 2,605
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265
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Automorphic Banach spaces
A Banach space $X$ is called automorphic if for every closed subspace $Y\subseteq X$ with $\dim X/Y=\infty$, every automorphism (= linear continuous isomorphism) of $Y$ can be extended to an ...
- 4,535
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204
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Given a totally ordered system of Banach spaces, can we we always change the norms to get isometric embeddings?
Given a real vector space $V$ which is the union of a totally ordered family of vector subspaces $V=\bigcup_{i\in I} V_i$. By that I mean that we assume that $(I,\leq)$ is a totally ordered set and ...
- 609
5
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103
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What conditions guarantee differentiability of higher-order functions?
Sorry if my language is not precise.
Let $f_i:\mathbb{R}^m\to \mathbb{R}^n$ be some differentiable functions in the sense of auto-diff. How can you construct new functions $g(f_1,f_2,f_3,...)$ such ...
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198
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If $\lim_{t\to +\infty} \int_{0}^{\pi} f(x)\exp(e^{xt}) \, dx=0$ then $f=0$ a.e?
Question, Let $f \in L^1(\alpha, \beta)$ , $\beta>0$ and
$$
F(x)= \int_{\alpha}^{\beta} f(t)\exp(e^{xt}) \, dt
$$ such that $\displaystyle \lim_{x\to +\infty}F(x)=0$. Does this imply that $f$ is ...
- 1,503
5
votes
0
answers
135
views
Criteria for tightness of Gaussian measures on Banach spaces
In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
- 505
5
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0
answers
255
views
Equality from the Grothendieck inequality
I asked the following question on math.stackexchange.com but have not received any response. So I would like to try my luck here.
This question is related to the Grothendieck inequality. Let field $\...
- 2,239
5
votes
0
answers
235
views
Proof of density of smooth functions in $H(\operatorname{curl})$ with $L^2$ tangential trace
I am trying to understand the proof of the following statement that is presented in the book “Finite Element Methods for Maxwell's Equations” by Peter Monk. The original source of the proof is a 1997 ...
5
votes
0
answers
116
views
Multiplier algebra of Fock space
For any vector space, one may form the tensor algebra with multiplication being the tensor product. For a Hilbert space $\mathcal{H}$, the analogous construction is the Fock space
$$
\mathcal{F}(\...
- 439
5
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0
answers
206
views
Equality of weak solutions for inner products inducing equivalent norms
This is a repost of a now-deleted MSE question that did not get any comments or answers.
$\textbf{Background}$: This question is mainly about two basic questions: How do we systematically obtain the ...
- 153
5
votes
1
answer
630
views
Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
- 10.5k
5
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169
views
Is the Grassmannian of a Banach space an infinite dimensional manifold?
Grassmannian of complemented subspaces in a Banach space is a Banach manifold. This is explained for example in the thesis of Douady and is rather analogous to the finite-dimensional case.
I would ...
- 344
5
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answers
74
views
Concentration bound on additive functions with constraints
Given a family of sets $F \subseteq P(\{1,\ldots,n\})$. I define the function $f_F:[0,1]^n \rightarrow R$ to be $f_F(x_1,\ldots,x_n)= \max_{S \in F} \sum_{j \in S} x_j$.
Given a series of independent ...
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5
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answers
208
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Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
5
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answers
280
views
Completeness of the space $L^p$ and the Axiom of Countable Choice
I am thinking about the proof that the usual $L^p$ spaces are complete.
So, let $(X,\mathcal{F},\mu)$ be a measure space and let
$p\in[1,+\infty)$.
Important: by a measure I mean a nonnegative $\sigma$...
- 935
5
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0
answers
137
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A concrete description of the projective tensor product of Lipschitz spaces
$\newcommand{\projtenprod}[2]{#1 \; \hat\otimes_\pi #2}$
$\DeclareMathOperator\Lip{Lip}\DeclareMathOperator\AE{AE}$
$\newcommand{\norm}[1]{\| #1\|}$
$\newcommand{\abs}[1]{| #1|}$
Background
...
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5
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answers
186
views
Is a Banach space with a predual of arbitrary order necessarily reflexive?
Let $X$ be a Banach space, by a predual of order $n$ ($n$ is a positive integer), I mean a Banach space $X_{n}$, such that the $n$-th dual $X_n^{(n)}$ of $X_n$ is isometrically isomorphic to $X$. My ...
- 1,586
5
votes
0
answers
158
views
Bochner–Minlos Theorem for locally convex spaces and their duals
Let $(X,\tau)$ be a locally convex space and $(X^{*},\tau_{s})$ be its topological dual space equipped with the strong topology. Denote by $S(X,X^{*})$ the collection of operators from $X$ to $X^{*}$ ...
- 3,515
5
votes
0
answers
315
views
Schauder basis in the Arens-Eells space
Context
Arens-Eells space. Let $X$ be a separable pointed metric space with base point $e$. An elementary molecule is defined as follows (Nik Weaver, Lipschitz Algebras, 2nd ed.)
$$
m_{pq} := \delta_p ...
- 437
5
votes
0
answers
211
views
Weaker analogues of amenability for groups of piecewise projective homeomorphisms
Let $A$ be a subring of ${\bf R}$ and let $H(A)$ be the group defined/constructed in Monod's 2013 PNAS paper. Monod showed that provided $A\neq {\bf Z}$, $H(A)$ is non-amenable. (The proof breaks down ...
- 25.8k
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votes
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145
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Second dual $X^{**}$ of ternary $C^*$-ring $X$ is again ternary $C^*$-ring?
Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
- 1,115
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163
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Commutator of pseudodifferential operator and multiplication operator
Cross-post from math.sx.
Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the ...
- 245
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0
answers
121
views
Which operations commute with fractional translation?
Let $\mathbf{v}$ be a real signal (i.e., an infinitely long vector).
A translation operation $T_{s}\mathbf{v}$ with integer $s$ is trivially
defined by $\forall i,s:\left(T_{s}\mathbf{v}\right)_{i}=v_{...
- 968
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votes
0
answers
199
views
Standard function spaces with the approximation property
A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...
- 313
5
votes
0
answers
419
views
Nonlinear variation of constants formula
Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
- 51
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votes
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100
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What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?
Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence.
Question: What is a minimal topological condition on $X$ ...
- 315
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218
views
When is it true that $Z(A)^{**} = Z(A^{**})$ for a C*-algebra $A$?
For a (unital) C$^*$-algebra $A$ with centre $Z(A)$ the bidual $A^{**}$ is a von Neumann algebra with centre $Z(A^{**})$ for which I believe that $Z(A)^{**} \subset Z(A^{**})$ holds by extending the ...
- 633
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0
answers
118
views
Good (Sidon) Approximation of "Bumps"
Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...
- 113
5
votes
0
answers
489
views
Dual norm for weighted Sobolev space
Consider a "weight" function $\rho: \mathbb{R}^{d} \rightarrow \mathbb{R}$ and the following weighted Sobolev norm:
\begin{equation}
\|h\|_{\dot{H}^{1}(\rho;\mathbb{R}^{d})}:=(\int_{\mathbb{...
- 407
5
votes
0
answers
278
views
Reflexive norm-closed subalgebras of $B(X)$
Let $X$ be a reflexive Banach space, and let $B(X)$ denote the set of all bounded linear operators $X\to X$.
Does there exist a norm-closed subalgebra $A\subseteq B(X)$ with the following properties?
...
- 2,605
5
votes
0
answers
156
views
Homotopy, contraction mapping and the inverse function theorem on Banach spaces
We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / ...
- 179
5
votes
1
answer
425
views
Positiveness of the largest Lyapunov exponent
Let $\alpha\in \mathbb{R} / \mathbb{Q}$, let $A(x)$ be the $2$-by-$2$ matrix
$$
A(x)=\begin{pmatrix}
\dfrac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\dfrac{1}{{\lambda}} \\
\dfrac{...
- 131
5
votes
0
answers
222
views
Right derived contravariant hom-functor
I am interested in additive categories appearing in functional analysis, in particular, the category $LCS$ of locally convex spaces and continuous linear functions. This category is not abelian but ...
- 16.4k
5
votes
0
answers
61
views
Minimizers of variational problems with less symmetry
In am interested in understanding the so-called Hartree ground states, namely, minimizers of variational problems of the form
$$\inf_{\phi\in H^1,~\|\phi\|_2=1}\left\{\int|\nabla\phi(x)|^2dx
-\int|\...
- 891
5
votes
0
answers
144
views
Is there a discrete Schrödinger operator with empty spectrum?
A relatively well-known example of (continuous) Schrödinger operator with empty spectrum is the complex Airy operator on the line, i.e., the operator acting on $L^{2}(\mathbb{R})$ given by the ...
- 2,188
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votes
0
answers
203
views
Global analysis on punctured surfaces
Global analysis on open manifolds seems pretty hard. For one, the space of $C^{n,\alpha}$ functions on an open manifold need not be a tame Fréchet space (see the post Are smooth functions tame? for ...
- 163