Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,435 questions with no upvoted or accepted answers
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Prove that this integral operator is onto
Let us consider the operator $T$ defined by$$\eqalign{
& T:{L^2}((a,b) \times (c,d)) \to {L^2}((c,d)) \cr
& Tf(s,x) \mapsto \int\limits_{q(x)}^{p(x)} {f(\alpha (s,x),s)ds} \cr} $$
where ...
- 617
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251
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Links between differing notions of "pseudo-measure"'; or, why that name?
(A pet peeve of mine is Mathematicians from field X noticing that field Y uses terminology which is very close to that from field X, and assuming there are Mathematical links. This question might be ...
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169
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A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
- 434
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2k
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On weak compactness of the unit ball in a reflexive Banach space
It is a well known result in functional analysis that a Banach space $X$ is reflexive if and only if the unit ball is weakly compact (compact in the weak topology). This result is also known as ...
- 364
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70
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Infinite trees whose spectrum has more than 3 connected components
I was wondering whether there exists any infinite tree $T$ such that the action of $\mathit{Aut}(T)$ on the set of vertices $V=V(T)$ has finitely many orbits, and whose spectrum $\sigma(T)$ has ...
2
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Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?
Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...
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Weak Hessian for solutions to certain quasilinear elliptic PDEs
In Chapter 4 of their famous treatise Linear and quasilinear elliptic equations, Ladyzhenskaya and Uraltseva deal with equations of the form
$$\sum_{i=1}^n\frac{d}{dx_i}a_i(x,u,\nabla u)+a(x,u,\nabla ...
- 3,146
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77
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Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
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194
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Extension of universal approximation theorem
Let $I_d:=[0,1]^d$ with $d\ge 2$. Define $C(I_d):=\{F: I_d\to\mathbb R \mbox{ is continuous}\}$ and
$$N(I_d):=\{F\in C(I_d): F(x)=\sum_{k=1}^n f_k(v_k\cdot x), \mbox{ where } n\ge 1 \mbox{ and } f_1,\...
user128095
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Lyapounov's inequality for Orlicz norms
When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$.
This inequality is a special (or rather limit) case of ...
- 4,432
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A holomorphic map into a Hilbert space with prescribed orthogonality
This is a variation of my previous question.
Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...
- 5,529
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Equivalent definition of real interpolation space
In one of his papers (On the Nash-Moser implicit function theorem, Ann. Acad. Sci. Fenn., Ser. A I, Math. 10, 255-259 (1985).), Lars Hörmander introduces a class of function spaces that seems related ...
- 1,263
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53
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References for linear relations on Hilbert spaces
I am trying to find a reference for linear relations (multivalued operators). I would like to have something which gives an introductory overview.
All I have found so far doesn't seem right for ...
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Principal symbol of a non-local operator and Atiyah–Singer index formula
I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
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Algebraic language of fundamental results in operator algebras
Ignoring topological structures of von Neumann algebras, the larger category of Baer $*$-rings was emerged. In the unique text written by Sterling K. Berberian (1), some concepts and results in von ...
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171
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How to use Stein-Tomas theorem to check to following inequality?
Recently, I am reading Rodnianski & Schlag
Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. In lemma 3.2, R&S said that by using Stein-Tomas theorem ...
- 429
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136
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Reference of a modified version of Hahn-Banach Theorem
I need a reference on where to find a modified version of Hahn-Banach Theorem that ensures that if $F(g) = \int g d\mu$ defines a positive linear functional on a subspace of $C_{0}(M, \mathbb{R})$, $F$...
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Are almost all continuous functions nowhere differentiable?
The question is mostly in the title, but to be precise, the space is $E={\cal C}^0( [0,1],\mathbb R)$, and 1) is there a reasonably natural measure on $E$ (at least invariant by translation, say) 2) ...
- 3,630
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165
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Run-away Volterra operator
For a continuous function $k:[0,1]^2\to \mathbb{R}$, let $A$ be the generalized Volterra integral operator on $C([0,1],\mathbb{R})$ defined by
$$
A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds, \quad t \in [...
- 5,405
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Closure of the space of holomorphic functions on the open disk in $\mathbb{C}$ with respect to a Hardy-space-like semi-norm
Let $\mathcal{A}\left(\mathbb{D}\right)$ denote the vector space over $\mathbb{C}$ of all holomorphic functions $f:\mathbb{D}\rightarrow\mathbb{C}$. Define the following semi-norm:$$\left\Vert f\right\...
- 1,284
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Some questions about convergence
I'm getting lost in the first couple of sentences of a paper by Cafarelli & Alt. The sections are organized as follows, 1.1 Data, 1.2 The problem, 1.3 the statement and relevant part of proof.
1.1 ...
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62
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Existence and uniqueness for semilinear problem
Consider the following problem:
$$-\Delta u + [(u)^+]^\alpha = 0,$$
where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
user139844
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Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?
Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...
- 369
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87
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Smooth structure for probability measures on separable Hilbert space
If $H$ is a separable Hilbert space (say, $L^2(\mathbb{R}^d)$ for concreteness), then it is well-known that the unit sphere $S$ of $H$ is a Hilbert manifold modeled on $H$ itself. The coordinate ...
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445
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Lax Milgram for non coercive problem?
I obtained the variational form of my problem. and the bilinear form is below.
Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have
$$a(u,v)=\int_\Omega u(t)...
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241
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Monotone approximation of continuous function by Lipschitz function
Let $X$ be a normed space, and $f\in C_b(X,\mathbb{R})$. How can I construct a sequence $\varphi_k,\,k\geq 1$ of bounded Lipschitz functions with $\varphi_k\downarrow f$ pointwise? Thanks
- 139
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154
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Browder's Fixed Point Theorem in uniformly convex Banach spaces with non-identical image
This is in fact an exercise from Dirk Werner's book "Funktionalanalysis", but I do think that the result is quite interesting and up to now, I can only partly solve this problem. From the point of my ...
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137
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Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
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Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces
In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
- 5,405
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Regularity for Laplacian operator on non-compact manifold
Let $(M,g)$ be a complete non-compact Riemannian manifold .
Thanks to @EveryLT, we know that the Poisson equation
$$\Delta u=f,$$
is solvable
for some $f\in L^2_k(M)$.
Q Suppose that $(M,g)$ is ...
- 1,915
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Small energy implies a lifting $\rho e^{i\theta}$
Set $T^N$ the $N$-dimensional torus and $u\in H^1(T^N,\mathbb{C})$. Can I say that if the energy$$\int_{T^N}|\nabla u|^2 +\frac12\int_{T^N}[1-|u|^2]^2$$ is small enough (let say lower than some $\...
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Could we assume without loss of generality that all coefficients are positive?
Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...
- 2,118
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95
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Exp-decay estimate of Schrodinger equation
Consider the equation $Hu=0$ with $u\in L^2(\Omega)$, where $H=-\Delta+V$ for some bounded continuous function $V$ and $\Omega$ is an un-bounded domain(e.g. $\mathbb R^n$). If $0$ is in discrete ...
- 1,915
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275
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Gagliardo-Nirenberg inequality for periodic functions?
I am interested in a Gagliardo-Nirenberg type inequality for functions in the space
$$H^1_T(\mathbb{R}^n)=\lbrace \psi=u+iv\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)\rbrace$$
...
- 383
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Time derivative in parabolic Hölder spaces
Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$.
Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...
- 301
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551
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Euler-Lagrange equations on a differentiable manifold
I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
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240
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Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
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A new topology on the dual of a locally convex space?
Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...
- 42k
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205
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almost magic unitary
A magic unitary is a unitary matrix $u=(p_{ij})_{ij}$ whose entries are all projections (in some Hilbert space) and in each row they sum to the identity and same holds for each column $(\sum_i p_{ij}=...
- 21
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60
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direct limit in locally convex modules and continuous map
Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...
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Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?
Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following:
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
- 839
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73
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Alberti rank-one theorem and irregular jump discontinuities
Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
- 839
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73
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Projection of BV function
Let $u \in [BV(\mathbb R^N)]^N$. We have
$$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$
where $a,b \in \mathbb S^{N-1}$.
What is the projection of $D^{jump}u$ in the direction $a$? And how can ...
- 839
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124
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Solving Fredholm integral equation in Lp
I have a very simple integral equation
$$
f(x) - \lambda \int_a^be^{x-y}f(y)dy=1
$$
which is Fredholm of the 2nd kind, with separable kernel. It is needed to find the values of $\lambda\in\mathbb R$, ...
2
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151
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A Banach or $C^*$ algebraic analogy of a classical fact in real analysis
Let $A$ be a commutative unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$.
Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$.
Assume that for every $\phi\in \...
- 366
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116
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Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 993
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0
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77
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How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
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125
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Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
- 366
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0
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119
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Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
- 21
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115
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What is the suitable tensor product for Holder spaces
We know that for $X\subset\mathbb R^m,Y\subset\mathbb R^n$ open, then $C^0(\bar X\times\bar Y)=C^0(\bar X)\hat\otimes_\varepsilon C^0(\bar Y)$ where $V\hat\otimes_\varepsilon W$ is the injective ...
- 938