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Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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Notation for a sum of Hilbert spaces

Let $\mathcal{T}$ and $\mathcal{P}$ be a finite- and an infinite dimensional Hilbert spaces. Does the notation $\mathcal{T} + \mathcal{P}$ stand for \begin{equation} \mathcal{S}=\mathcal{T} + \...
Vuk's user avatar
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Numerical radius norm is equal to the operator norm

Currently, I am studying numerical ranges and numerical radius of linear operators. As one of the references, I am studying the book ``Numerical Range: The Field of Values of Linear Operators and ...
Tutun's user avatar
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Lipschitz continuity and absolute continuity [duplicate]

Given an interval $I\subset \mathbb{R}$, a function $f:I \to \mathbb{R}$ is said to be absolutely continuous if for all $\epsilon >0$ exists $\delta>0$ s.t $\sum_{i=1}^N |f(c_i)-f(d_i)| \le \...
Shiva's user avatar
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Convergence in Total Variation vs implies Uniform Convergence

Let $(f_n)_{n\in \mathbb{N}}$ and $f$ real function defined on an interval $[a,b]$ s.t. $f, f_n$ are bounded variation function on $I$, and suppose that $TV(f_n) \to TV(f)$. I should prove that it ...
Shiva's user avatar
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Stone-Weierstrass theorem proof in Rudin analysis

In chapter 7 of Rudin analysis in question number 23 we have: Put $P_0 = 0$, and define, for $n = 0, 1, 2, \ldots ,$ $$P_{n+1} = P_n + \frac{ (x^2 - P_n^2) }{2}$$ Prove that $ \lim_{n\to ∞ } P_n(x) = ...
arvin asadi's user avatar
2 votes
2 answers
39 views

Bounded operators on the span of a set of vectors

Let $E$ be a Banach space and $S$ a subset of vectors in $E$. Suppose that a linear operator $T$ satisfies the property $$\|Tx\| \leq C \|x\| \text{ for all }x \in S.$$ Does this imply that $T$ is ...
pseudo-goldstone's user avatar
1 vote
1 answer
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Dimension of a linear subspace invariant under closure.

Let $X$ be an infinite dimensional normed vector space over $\mathbb R$ and $W \subseteq X$ an $m$-dimensional subspace. Is it true that the topological closure $\overline{W} \subseteq$ is also an $m$-...
azuwaterloo's user avatar
1 vote
1 answer
38 views

Prove that $\Sigma_A(\overline A)\leq\operatorname{hd}(A,\overline A)\leq\operatorname{md}(A, \overline A)$.

Let $A\in \mathbb C^{n,n}$ and perturbed matrix $\overline{A}\in \mathbb C^{n,n}.$ Suppose spectrum of A, $\sigma(A)=\{ \lambda_1,\lambda_2,...,\lambda_n\}$ and $\sigma(\overline A)=\{ \overline\...
Unknown x's user avatar
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Helmholtz - Hodge decomposition in H(curl)

I'm looking for Helmholtz-Hodge type decompositions but for vector fields slighty more regular than $L^2(D,\mathbb{R}^3)$. I'm familiar with the results in the books of Lions and was wondering if ...
Caillou's user avatar
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Equality of measures on functional spaces

It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
Anico's user avatar
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1 answer
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condition of banach space in Proposition 3.13 brezis

Let $\sigma(E^*,E)$ the weak-* topology in $E^*$ and $(f_n) \in E^*$ a sequence. Then If $f_n \rightharpoonup^* f \implies (||f_n||)$ bounded, and $||f|| \leq \liminf |||f_n||$. we can use that $f_n(x)...
Franco Gómez's user avatar
1 vote
1 answer
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Spectre map is continuous under hausdorff-metric

could anyone provide me a hint on how to prove the following: Let X be a Banach space and $d_H$ be the Hausdorff metric defined on the compact subsets of $\mathbb{C}$, $$ d_H(M,N):=\max ( \sup_{x \in ...
IMM's user avatar
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1 answer
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Singular integral kernel

Define $$ I : L^2(\mathbb R) \to L^2(\mathbb R), (If)(y) = \int_0^\infty \frac{f(x+y)}{\sqrt{x}} dx. $$ Why is this well-defined and a bounded operator? Since $\int_1^\infty \frac 1 {\sqrt{x}} dx$ ...
univalence's user avatar
1 vote
2 answers
87 views

Prove that : $\rho(x,y)= \sqrt{(x_1-y_1)^2+4(x_2-y_2)^2} $ is a metric

let $X= R^2$. If $x$ and $y$ are points in the plane and their cordinates are $x=(x_1,x_2), y=(y_1,y_2)$ prove that the given function is a metric. I am currently stuck on proving the triangular ...
Marco Marino's user avatar
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Convergence of the derivative of a BV function in sense of measures [closed]

Suppose I have a smooth sequence $f_{n} : \mathbb{R} \to \mathbb{R}$ with $f_{n} \to f$ strongly in $L^{1}_{loc}(\mathbb{R})$ and $f \in BV_{loc}(\mathbb{R})$. Is there any way to justify that $\...
duelspace's user avatar
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Berge's Maximum Theorem for Parameters that are Functions

I am given some functions $f_i$ that are $C^1$, $f_i : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ and some budget $z \in \mathbb{R}_+$. I have some constrained optimization problem with a continuous ...
Ator's user avatar
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Support of the Fractional Stationary Ornstein Uhlenbeck process (first kind)

I am interested in the support of the fractional stationary Ornstein Uhlenbeck process (first kind). In particular, I want to know whether smooth functions (or even only smooth functions starting at 0)...
Chris's user avatar
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Implicit non linear formula to explicit polynomial formula.

Suppose that $\{P_n\}$ are sequential functions of polynomials of degree $n$, and $x$ is between $[-1,1]$. Put $P_0 = 0$, and define, for $n = 0, 1, 2, \ldots ,$ $$P_{n+1} = P_n + \frac{ (x^2 - P_n^2)...
arvin asadi's user avatar
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1 answer
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For two positive linear functional $\phi$ and $\psi$, if $\phi \leq \psi$, what is the relation between the two GNS representation?

The above question is singled out from M. Izumi's paper. The precise reference is pp 102 in Inclusions of simple $C^*$-algebras in Crelle. In fact, I don't see why two positive functionals $\phi$ and ...
Squidgame's user avatar
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Dilation of Open Ball to Unit Ball in Complex Plane

Suppose $S=\{z\in \mathbb{C} : |z-a|<r \}$ is an open unit ball in right half plane. Then can we get $c>0$ such that $c. S = \{cz : z\in S \}\subset \{z\in \mathbb{C}: |z-1|<1$? I tried the ...
VINI's user avatar
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0 answers
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$f = g$ in $H^{-1}$ implies $f = g$ a.e.?

I have two functions $f, g \in H^1_0$. But I could only show $|f - g|_{H^{-1}} = 0$. Does that imply $f = g$ almost everywhere? I think this should follow immediately since if $f = g$ in $H^{-1}$, ...
mathdoge's user avatar
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1 answer
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prove that $\|f\|=1$, where $f : X\to X/\mathscr M$ is the projection

We define $$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$ Let $X$ be a normed vector space and let $\mathscr M \subset X $ be a closed subspace. Define $$X/\mathscr M :=\{...
A12345's user avatar
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Measurable in one variable and regular in the other

Suppose $(X,\mathcal{B},\mu)$ is a measure space and for $y\in\mathbb{R}$ suppose that the function $f(\cdot,y):X\to X$ is measurable and $f(x,\cdot)$ is continuous. In many cases, I have seen that ...
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Every non zero element of a $C^{\ast}$-algebra is regular?

Let $\mathcal{A}$ be a $C^{\ast}$-algebra and $a \in \mathcal{A}$. We say $a$ is regular if there exists $b \in \mathcal{A}$ such that $a=aba$. It is clear that every invertible element of $\mathcal{A}...
Math Lover's user avatar
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How to approach problems about stronger norms? [closed]

For a vector space $X$ and two norms $\lVert \cdot \rVert$ and $\lVert \cdot \tilde{\rVert}$ we say that $\lVert \cdot \tilde{\rVert}$ is stronger than $\lVert \cdot \rVert$, if there exists a ...
Philip's user avatar
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2 votes
0 answers
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Proof that the hausdorff distance $d_H( \sigma (A), \sigma(B)) \le r(A-B)$ for commuting operators $A,B$

I am supposed to prove that for two bounded commuting operators $A, B$ on a banach space, the hausdorff distance of the spectra $d_H(\sigma(A),\sigma(B))$ is less than or equal to the spectral radius ...
Julius Himmel's user avatar
2 votes
1 answer
104 views

Showing $T$ is a compact operator

Let $\{\varphi_n\}$ be an orthonormal sequence in a Hilbert space $H$ and consider the operator $T:H \to H$ defined by $$T(f) = \sum_{n=1}^\infty a_n \color{blue}{\langle \varphi_n, f \rangle} \...
Grigor Hakobyan's user avatar
2 votes
1 answer
69 views

The norm of the operator cos(A)

The operator $A$ is defined as $A(x,y) = (\frac{-3π}{4} x + \frac{π}{2} y, \frac{π}{2} x)$. I need to find the norm of the $cos(A)$ operator. I tried writing $cos(A)$ as a series in the hope that at ...
Tom Sawyer's user avatar
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0 answers
45 views

Interaction of pullback and the Fourier transform

I'd like to understand how the spectra of functions on a given domain are affected by (different kinds of) maps of that domain. Specifically, consider the Schwartz space $S(\mathbb R^n)$ of test ...
Yaque's user avatar
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1 answer
44 views

Dual of weak topologies

The following question is a follow up to an earlier question I asked (Isometric isomorphism and annihilators of annihilators). So, here is the question: Let $(X,{\cal T}_{||\cdot||})$ be a normed ...
prgnts123's user avatar
1 vote
0 answers
17 views

Parabolic maximum principle on the whole space

I am aware of the following maximum principle for the heat equation: Let $T \in (0, \infty]$ and $u$ satisfying \begin{align} \partial_t u - \Delta u = 0 &\quad \text{on }\mathbb R^n \times (0, T]...
Falcon's user avatar
  • 4,072
0 votes
1 answer
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An explicit formula for orthogonal functions

I am interested in orthogonal functions for the inner product $$\int_a^b f(x)g(x) \alpha(x) dx$$ where $\alpha$ is a non-negative function. Given linearly independant functions $f_0, \ldots, f_\ell$, ...
Wirdspan's user avatar
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1 vote
0 answers
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When is the closable composition of operators affiliated to a von Neumann algebra also an affiliated operator?

Let $S$ and $T$ be closed (unbounded) operators affiliated to a von Neumann algebra $M$. Can we say anything about when the closure of their composition $T \circ S$ is also affiliated to $M$ (assuming ...
szantag's user avatar
  • 81
3 votes
1 answer
52 views

MacCluer Exercise 3.2

In MacCluer's Elementary Functional Analysis, exercise 3.2 states: We introduce some terminology for the purpose of this problem: If $X$ is either a real or complex vector space (meaning that the ...
Jeffrey Chen's user avatar
1 vote
0 answers
70 views

Finding eigenvalues and eigenfunctions of integral operator

I have the following problem: Let $A:C^\infty\rightarrow C^\infty$ with $$(Af)(x)=\int_{0}^{1}|x-y|f(y) dy.$$ I need to find the eigenvalues and eigenfunctions of $A$. To find the eigenvalues and ...
Fluadl's user avatar
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2 votes
1 answer
24 views

Isolated Eigenvalues of Finite Multiplicity in the Spectrum of a Self-Adjoint Operator with Compact Perturbation

I am studying spectral properties of operators, specifically working through Ronald G. Douglas' "Banach Algebra Techniques in Operator Theory." While doing exercise 5.15, I encountered the ...
f yz's user avatar
  • 21
1 vote
1 answer
86 views

Prove that $\|f\|=n^{1/2} $

We define $$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$ We said that $f$ is linear bounded function if $$ \exists M>0 : \forall x\in X, \|f(x)\| \le M\|x\| $$ Also, ...
A12345's user avatar
  • 131
3 votes
1 answer
29 views

Operator with basis-independent trace but which is not trace class

Let $A$ be a trace-class operator on a separable Hilbert space $H$. By definition, this means that the quantity $$\sum_i\langle|A|e_i,e_i\rangle$$ is finite, where $\{e_i\}_i$ is an orthonormal basis ...
geometricK's user avatar
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2 votes
1 answer
70 views

Two sigma-algebras on $l^\infty$

Let $\mathcal{B}_w(l^\infty)$ denote the Borel $\sigma$-algebra generated by the weak topology on $l^\infty$. Let $Cyl(l^\infty)$ denote the cylindrical $\sigma$-algebra, that is, it is the $\sigma$-...
Zlyp's user avatar
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1 vote
1 answer
32 views

Let $A:=\{f\in C^1(\mathbb{R}): f, f'\in L^1(\mathbb{R}\}$. Then are the Schwartz functions dense in $A$ w.r.t. $\|f\|=\|f\|_1+\|f'\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): f, f'\in L^1(\mathbb{R}\}$. Then is it true that Schwartz functions are dense in $A$ with the norm $\|f\|=\|f\|_1+\|f'\|_1$.? My guess is the above statement shoud be ...
CCCC's user avatar
  • 113
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0 answers
15 views

How does one prove that a given function is localy summable? [closed]

For example the function ln(x+iy)
Mirindra Fandresena's user avatar
1 vote
0 answers
24 views

Open mapping theorem for spaces of holomorphic functions

In studying several complex variables, I came across an exercise in what is effectively an application of the open mapping theorem for Frechet spaces without actually using said theorem. The situation ...
Maths Matador's user avatar
3 votes
0 answers
39 views

The scaling effect on the Besov norm

The Besov spaces can be defined using the dyadic decomposition. For this purpose, let $\varphi \in C^\infty_c(\mathbb R^n)$ be such $ {\rm supp}(\varphi) \subseteq \left\{x; \, \frac 12< |x| &...
A. PI's user avatar
  • 639
1 vote
2 answers
25 views

bv space is a direct sum of $bv_0$ with a one dimensional subspace

We define $bv=\{x=\{x_n\} | \sum_{k=1}^\infty |x_{k+1} -x_k| < \infty , x_k \in \mathbb C\} $ $C_0=\{x=\{x_n\} | lim x_n = 0,x_k \in \mathbb C\} $ And $bv_0=bv \cap c_0$ I need to prove that bv ...
A12345's user avatar
  • 131
0 votes
0 answers
16 views

Understanding the transpose of a (generalized) Calderon-Zygmund operator

I have been reading Fourier Analysis by Javier Duoandikoetxea, and I have a question about generalized Calderon-Zygmund operators. For reference, I state the definition as given in the book. ...
Aniruddha Deshmukh's user avatar
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0 answers
91 views

Approximating self-maps of $[0,1]$

Let $\mathrm{Inc}([0,1])$ denote the space of continuous, increasing functions $f:[0,1]\rightarrow [0,1]$ such that $f(0)=0$ and $f(1)=1$. I want to find a countable family of functions $f_n\in \...
Alvaro Martinez's user avatar
4 votes
2 answers
74 views

Analysing a Lebesgue integral inequality for $|t^{-n} \phi(x/t)|$, where $\phi \in C_c^\infty \cap L^1$ with $\| \phi \|_1 = 1$.

Context. Let $C^k(\mathbb R^n)$ denote the space of functions defined on $\mathbb R^n$ that are $k$ times continuously differentiable, where $k \geqslant 1$ is an integer. As usual, define $C^\infty(\...
xyz's user avatar
  • 1,079
0 votes
0 answers
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The $L^\infty$ boundedness of the resolvent of Harmonic Oscillator in terms of seminorms

I am reading Watanabe's book "Lectures on Stochastic Differential Equations and Malliavin Calculus". In Page 48, it is said that by means of (damped) harmonic oscillator $1+|x|^2-\Delta$, we ...
ze min jiang's user avatar
1 vote
0 answers
18 views

Unboundedness of Differential Operator by Fourier Transformation of Multiplication Operator $||x|| = \infty, D \mapsto k \implies ||D|| = \infty ?$

The multiplicative operator $T_g$ is defined by $f(x) \mapsto g(x)f(x) $ and is bounded in $L^2[\mathbb{R}]$ if $||g(X)||_\infty < \infty$. The operator $T_x$ is unbounded. The operator $D:=-i\...
theta_phi's user avatar
  • 115
1 vote
1 answer
55 views

If $\phi \in C_c^\infty(\mathbb R^n) \cap L^1(\mathbb R^n)$ such that $\| \phi \|_1 = 1$, then the dillations $\phi_t$ preserve these properties?

Context. Consider the usual Lebesgue measure on $\mathbb R^n$ and denote by $L^1(\mathbb R^n)$ the usual space of measurable functions that are integrable. Moreover, let $C^k(\mathbb R^n)$ denote the ...
xyz's user avatar
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