# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

6,741
questions

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28 views

### Eigenvalues associated to differential opertator over natural spline spaces

Let $S_n^k$ denote de linear space of natural splines of degree $k$ and uniform grid $a < t_1 < \ldots < t_n < b$ over the bounded interval $[a,b] \subseteq \mathbb R$.
For any $m \in \...

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**1**answer

462 views

### Analytic functions where all derivatives vanish at infinity and which are bounded

Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$.
I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...

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61 views

### Non-linear weak*-continuous left inverses

Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\...

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61 views

### A question in Sobolev spaces involving time

Let $X$ be a Banach space, we understand $L^1(0, T, X)$ is the space of strongly measurable functions from $[0, T]$ valued in $X$, that is integrable. Assume ${\bf u}\in L^1(0, T, X)$, we say ${\bf v}\...

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35 views

### Inverse associated Legendre transform of products

Debnath and Harrell 1976 introduced the associated Legendre transform of a function $F(x)$ and its inverse,
\begin{align}
f(n,m) &= T_n^m[F(x)] = \int_{-1}^1 (1-x^2)^{-m/2} P_n^m(x) F(x) dx \\
F(x)...

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112 views

### Hardy Littlewood maximal function bounds

Let $u \in W^{1,p}(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$ be a given function for some $1<p< \infty$ and let $k \in \mathbb{R}$ be any number and consider the following maximal function
$$
...

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89 views

### Spectrum of a self-adjoint operator and spectral measures

Let $T$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$, with spectrum $\sigma(T)$. For any $x,y\in \mathcal{H}$, denote by $\mu_{xy}$ the spectral measure of $T$ with respect to $x$ and $...

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223 views

### The contractivity of the heat semigroup in $L^p$ spaces

Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...

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424 views

### Wildly discontinuous linear functionals

Let $X$ be a Banach space, $H\subseteq X$ be a dense hyperplane, and $f$ be a continuous linear functional defined on $H$. Then $f$ is uniformly continuous and hence it admits a unique continuous ...

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53 views

### Definition question: asymptotic-$\ell_{p}$ versus coordinate-free asymptotic-$\ell_{p}$

Let $(e_{j})_{j=1}^{\infty}$ be a basis for the Banach space $X$. If there exist constants $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$,
\begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}...

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54 views

### Convergence in weak dual topology $\sigma(L^\infty, L^1)$

Let $f\in L^\infty(\mathbb{R})\cap C(\mathbb{R})$, that is $f$ is continuous and bounded on $\mathbb{R}$. Let $S_r$ denote the shift by $r\in \mathbb{R}$: $S_r f=f(\cdot-r)$.
Suppose $S_{r} f $ ...

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67 views

### Integration on quasi-Banach spaces and Schatten ideals

Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its ...

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97 views

### Example of linear functionals on $B(H)$

Let $H$ be a Hilbert space and $B(H)$ denotes the space of all continuous linear operators on $H$. I am looking for a class/example of bounded linear functionals $B(H)\to \mathbb C$ which cannot be ...

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539 views

### Arzelà-Ascoli theorem and Hölder spaces

Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...

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70 views

### $\ell^1$-bound on graph laplacian with weight

Consider the $\mathbb Z^2$ lattice, we then define for $u=(u_{ij})_{i,j \in \mathbb Z}$ the discrete Laplacian
$$(\Delta u)_{i,j}=u_{i+1,j}+u_{i-1,j}+ u_{i,j+1}+u_{i,j-1}$$
and the weight which pushes ...

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40 views

### Non-square multiplication operator matrix

Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$.
Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$
Intuitively, $$K: [L^2(0,1)]^n ...

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104 views

### Regarding the boundary of the range of a holomorphic map on the unit disc

Let $\Omega$ be a convex domain in $\mathbb{C}^n$. Let $f:\mathbb{D}\longrightarrow \bar\Omega$ be a holomorphic function. Let $z\in f(\mathbb{D})\cap \partial\Omega$. Let $\phi$ be a linear ...

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158 views

### Example when Kantorovich condition would not hold

Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator
$$
(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.
$$
Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...

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72 views

### Error bounds on the expansion of square root of matrix

I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...

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104 views

### Unique solution of a 1-D ODE with a bounded positive right-hand-side

Consider the initial value problem $$\dot x(t) = F(t,x), \quad t \in (0,T)$$ with given initial datum $$x(0) = x_0 \in \mathbb R.$$ More precisely we consider the integral equation $$x(t)=x(0)+\int_0^...

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84 views

### Ellipsoid in $L^p([0,1],\lambda)$ spaces?

Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known that for $ 1\leq p < q \leq +\infty$ we ...

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37 views

### Approximate identities on the unit disk and going beyond a power series' radius of convergence

Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. ...

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95 views

### Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]

Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$

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41 views

### Characterizing a set of functions

Let $\mathscr C$ be the space of measurable functions $f$ on the interval $[0,1]$ that can be written in the form
$$ f(t)=\sum_{k=0}^\infty c_k e^{-kt},$$
for some square summable series $\{c_k\}_{k=1}...

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102 views

### Embedding Hermitian forms into Hilbert spaces

Let $H$ be a nondegenerate, not positive definite, Hermitian form on a complex vector space $V$ such that
$$|H(x,y)|\le u(x)u(y)\tag{B}$$
for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ ...

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69 views

### On strong resolvent convergence of unbounded operators

Given a sequence of unbounded self-adjoint operators $\Delta_N$ defined on some fixed domain of the Hilbert space $L^2(\mathbb{R})$ converging to i*identity in the sense that: for all $\phi\in\...

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143 views

### “Mollification” of a convex function with a finite set of points unchanged

$f:\mathbb R\to\mathbb R$ is a convex continuous function. We have a finite or a countable set of triples: $\{(x_n,f(x_n),D_n)\}_{n\in N}$, where $D_n$ is the slope of a tangent line $L_n$ at $x_n$ (...

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142 views

### gap in a Banach spaces ultrapower proof

This is an adaptation of a Heinrich proof, but I'm missing a key ingredient.
Conjecture. Suppose $(x_n)_{n=1}^\infty$ is a Schauder basis for a Banach space $X$ whose canonical isometric copy in $X^{*...

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71 views

### Weak sequential compactness on the space of compact operators

Let $E,F$ be Banach spaces and let $A\subset K(E,F)$ be a subset of the space of compact operators from $E$ to $F$.
A result by Kalton states that $A$ is weakly compact if and only if $A$ is WOT* ...

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138 views

### Is a specific product function orthogonal to all harmonic functions

Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...

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52 views

### A function with a dense set of periods and two values, except for a countable set, is a constant a.e. with respect to Lebesgue measure

I am reading the book Vector Measures of Diestel and Uhl, especifically example 6 of Sierpinski in Chapter 2, about the construction of a function that is weak$^*$-measurable but not weakly measurable....

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214 views

### Can quotient space be isomorphically isometric to some closed subspace of original one?

Suppose $\mathcal{B}$ is a Banach space which is not isomorphic to a Hilbert space. What condition should I pose on $\mathcal{B}$ such that for every closed subspace $\mathcal{M}$, the quotient space $...

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53 views

### Basic question about convergence of top and penultimate eigenvalues of a sequence of operators

$\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\C}{\mathbf C}$
Questions
Let $I$ be the unit interval.
Let $H=L^2(I)$ and $T:H\to H$ be a compact self-adjoint ...

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69 views

### About the definition of lineal convexity

I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set $A\subset \mathbb{...

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64 views

### Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components

Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...

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186 views

### Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?

Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...

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152 views

### Operator norm and spectrum

I am wondering about when an operator norm coincides with the maximum eigenvalue of an operator and there is one particular aspect that confuses me quite a lot.
Let's say we have a symmetric positive ...

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79 views

### Determining what happens to the spectrum of Schrödinger operator as boundary condition changes

I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...

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45 views

### Order of ultradistribution

I know that the order of any distribution of compact support is finite. Is this true in the case of ultra distribution of compact support ( dual of Denjoy-Carleman space)?

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52 views

### Prove integral inequality for divergence-free vector fields

Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold?
$$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...

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147 views

### Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)

Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...

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99 views

### Definition of homogeneous Sobolev spaces

As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...

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45 views

### Multiobjective Optimization with (too many?) functions

Consider a multiobjective optimization problem $$\min\limits_{x\in \Omega} f(x),$$ where $f:\Bbb R^n \rightarrow \Bbb R^m$ and $\Omega \subseteq \Bbb R^m.$ A point $\bar{x} \in \Omega$ is said to be:
...

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140 views

### For self-adjoint $A$ and $B$, when is $(A+iB)^*$ the closure of $A-iB$?

Suppose that I have two self-adjoint operators $A$ and $B$ such that $\mathcal{D}(A)\cap\mathcal{D}(B)$ is dense and $B$ positive. Then $A\pm iB$ (with domains $\mathcal{D}(A)\cap\mathcal{D}(B)$) are ...

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105 views

### Conditions on the inequality with a gauge norm

Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...

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76 views

### A Riemann Hilbert problem on the unit square

Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...

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113 views

### Mixed partial derivatives of planar functions converging to delta distribution

Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...

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**1**answer

88 views

### Invertibility of neural network as operator on Wasserstein space

Question statement: Consider the space of probability measures with finite second moments $P_2(\mathbb{R}^d)$, which is equipped with the Wasserstein-2 distance $W_2$, and the square integrable ...

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54 views

### Ergodicity properties of a linear operator on the Wiener Space

Suppose I have a centered Gaussian measure $\mu$ on a separable Banach space $B$. Let $H \subset B$ be its Cameron-Martin space and assume that it is dense in $B$.
Suppose that $U:H \to H$ is a linear ...

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102 views

### Inflating the double dual of a C*-algebra (matrix algebra of double dual)

in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})\cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can't really see it. Actually, it is enough for me ...