Functional analysis is 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, as well as measure, ...

learn more… | top users | synonyms (1)

3
votes
1answer
71 views

Existence of a solution to $f(x) = \int_0^1 k(x,y) f(y) dy$

Let $X = (0,1)\times (0,1)$ with the Lebesgue measure, and $k\colon X \to \mathbb{R}$ be a measurable non-negative function such that $$ \int_0^1 k(x,y) dy = 1$$ for every $x \in (0,1)$. My question ...
2
votes
0answers
21 views

Normal transformation with eigenvalue in real and complex case

It is known that in finite unitary space, due to spectral theorem, for a normal transformation,if the eigenvalues are 1)real 2)positive 3)absolute value 1,then it is 1) self-adjoint 2)positive ...
0
votes
0answers
56 views

Comparison of Sobolev spaces on an open or closed interval

As noted in my previous question, I am currently working through some books on Sobolev spaces. I am struggling to determine whether, given an interval $I=(0,a)$,the Sobolev spaces $W^{m,p}(I)$ and ...
8
votes
1answer
107 views

Sufficient Condition for $f\in L^{1}(\mathbb{R}^{d})$ to belong to $L^{2}(\mathbb{R}^{d})$

Question. Let $\left\{\varphi_{j}\right\}$ be a complete orthonormal system for $L^{2}(\mathbb{R}^{d})$ such that each $\varphi_{j}\in C_{b}(\mathbb{R}^{d})$ (the space of continuous, bounded ...
2
votes
2answers
43 views

Examples of algebras that have a bounded approximate identity

We note that $L^{1}(\mathbb R)$ is an algebra with respect to convolution without identity element. However, regarded as a Banach algebra, $(L^{1}(\mathbb R), \ast) $ has a bounded approximate ...
1
vote
0answers
23 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

Let $\mathcal{L}$=space of all lines $L$. The X-ray transform is defined here: https://en.wikipedia.org/wiki/X-ray_transform $Xf:\mathcal{L}\to \mathbb{R}$ is defined by: $Xf(L)=\int_{t \in ...
0
votes
0answers
9 views

Variational function versus variational solution

I want to minimize the functional $F[f(x)]$ and I'm going to try this in two different ways: First I am going to numerically minimize the functional $F[f(x)]$, leading to the "true solution" $f(x)$. ...
0
votes
1answer
22 views

Completeness of a system: For all $n$ and within an interval?

Why is the system $\sin((2n-1)x)$ for $n=1,2,\cdots$ complete in $L^2[0,\frac\pi2]$? This means that the Euclidean norm converges for $n=1,2,\cdots$ and for all $x\in[0,\frac\pi2]$ How does one prove ...
0
votes
3answers
51 views

Function on $\mathbb Z^2$ whose value equals the average of values at adjacent points $\Rightarrow$ function is constant

This is a reference request. I am not asking for a proof. If I remember correctly, there is a theorem that states that if a bounded [criterion added after editing] function $f:\mathbb Z^2\to\mathbb ...
3
votes
2answers
29 views

A condition equivalent to equicontinuity

I am doing a problem which is an application of the Arzela-Ascoli theorem, which boils down to proving that a certain condition is equivalent to equicontinuity. Specifically, I am given a sequence ...
3
votes
1answer
65 views

Multiplication operators on $L^2$

Let $X$ be a $\sigma$-finite measure space, and let $g$ a measurable complex-valued function $X$, which lies in $L^\infty(X)$. I would like to determine sufficient and necessary properties for the ...
-1
votes
0answers
23 views

A is an Hilbert operator, $(A+I)^{-1}$ is continuous with dense domain then A is essentially self-adjoint

We have no information about A, just that is an operator defined in $X\subset H$ where $H$ is a Hilbert space.
4
votes
1answer
50 views

An mixed weak star convergence problem

Let $\Omega\subset \mathbb R^N$ open bounded. Given a sequence of Radon measure $(\mu_n)$ such that $\mu_n\to \mu$ in weak star sense in $\mathcal M_b(\Omega)$ and $\|\mu_n\|\nearrow \|\mu\|$. Also ...
1
vote
0answers
12 views

Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . On the other hand interpolation space which is defined in the wikipedia link: ...
2
votes
1answer
171 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
5
votes
0answers
170 views

Doubts relating to Spaces of type $\mathcal{S}$

I have doubts in the following two questions : 1) What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , ...
3
votes
1answer
47 views

Why is the image of a C*-Algebra complete?

I am currently working through the book by Bratteli and Robinson on C* and W* algebras, there is one point at the beginning of chapter 2.3 that is frustrating me. If we take *-morphism to be a ...
1
vote
1answer
36 views

Doubts regarding the upper bound for Total Variation

I was studying a chapter on Total Variation & Compactness, where I had gone through the following portion: " We can also relate the total variation with the shifted $L^{1}$-norm. Define: ...
1
vote
1answer
122 views

A matrix is positive if and only if it is Hermitian and its eigenvalues are positive [duplicate]

I want to show the equivalence of two definition of positivity. Let $A \in \mathcal{L}(H)$, where $\mathcal{H}$ is the $n-$dimensional Hilbert Space $\mathbb{C}^n$. $A$ is positive if $\langle x,Ax ...
2
votes
1answer
21 views

If the gradient of $f$ at $x$ has the same direction with $x$ for all $x$, is $f$ radial?

I would like to ask the following question: If $f:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}\rightarrow% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ is ...
1
vote
1answer
50 views

The approximation of supremum of a set.

Given $\Omega \subset \mathbb R^N $ be open and let $g$: $\Omega\to \mathbb R^+$ be a $l.s.c$ function such that $g\geq 1$, but not necessarily bounded above. Also assume that there exists a sequence ...
2
votes
3answers
76 views

Is such a multivariate function the product of two univariate functions?

Let $f: \mathbb{N}^2 \rightarrow \mathbb{R}$ be a function of two variables, $f=f(x,y)$. $f$ has the following property: $$ \sum_{y\in A} f(x,y) = 0 $$ where sum on $y$ runs over a fixed ...
3
votes
1answer
50 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let ...
2
votes
0answers
24 views

Amenability; topology on power sets

I am currently reading the proof of Proposition 2.2. (a direct limit of discrete amenable groups is amenable) in http://people.maths.ox.ac.uk/kar/amenable.pdf . The proof uses the Tychonoff theorem ...
1
vote
0answers
63 views

Theorem 6.28 of Rudin's Functional Analysis

I am trying for some time to prove theorem 6.28 of Functional Analysis by Rudin.The theorem says that if Ω is an open subset of $\mathbb{R}^n$ and $Λ\in D^{'}(Ω)$,then there is a family ...
0
votes
1answer
40 views

Unconditional basis in $c_0$

We know that in $c_0$ the standard unit vector basis $(e_i)_{i=1}^{\infty}$ is an unconditional basis. For $n\in\mathbb{N}$, let $s_n=\sum\limits_{i=1}^{n}e_i$, my question is that How to prove ...
-1
votes
4answers
47 views

On characteristic function

Let $X$ be a set and $A,B\subset X$. Can we consider $\mid\chi_A-\chi_B\mid$ as a characteristic function of some subset of $X$? If yes which subset?
1
vote
0answers
45 views

Subsequences and blocks of Schauder bases

Suppose $X$ is a Banach space and $(e_n)$ and $(f_n)$ are both Schauder bases of $X$. Does there exist a proper closed subspace $Y\subset X$, and appropriate subsequences of $(x_n)$ and $(y_n)$ that ...
0
votes
2answers
24 views

Complete eigen-vector basis from non invertible linear application

Consider a non-invertible linear application $O$ acting on a Hilbert space (quantum mechanics). Is there still any chance to find a complete basis of $O$ eigen-vectors or no?
-2
votes
2answers
48 views

What is meant by “functional analysis is the study of vector spaces endowed with a topology” [closed]

Lecture notes on Functional Analysis by Razvan Gelca open with the definition: Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces. ...
1
vote
1answer
36 views

A claim in Krengel's book on Ergodic Theorems.

In Krengel's it's argued that the fact that $\exists 0\ne u \in L_\infty$ orthogonal to $(zI-T)L_1$ , where $z$ is a complex number on the unit circle, $|z|=1$, then $T^* u = zu$. I don't understnad ...
0
votes
1answer
19 views

Direct Integral: Measurability

Given a Borel space $\Omega$. Consider plain functions: $$\eta,\vartheta\in\mathcal{F}(\Omega):=\{\eta:\Omega\to\mathbb{C}\}$$ The implication is wrong: ...
5
votes
2answers
75 views

Does orthogonal decomposition characterize direct sums in Hilbert space?

Let $H$ be a Hilbert space with inner product $(\cdot, \cdot)$. I know that if $M$ is a closed subspace of $H$, then $H$ can be written as the direct sum $M \oplus M^\perp$, where $M^\perp$ stands ...
4
votes
0answers
34 views

“Contradiction” to Bochner's theorem for distributions

I recently asked a question "For what values of $\lambda$ the distribution $(x-i\epsilon)^{\lambda}$ is positive?". User Marcel was kind enough to point out in his answer that one uses Bochner's ...
2
votes
1answer
49 views

Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
0
votes
0answers
29 views

Interpolation Inequality for almost everywhere differentiable functions?

For a function $f :\mathbb{R}\rightarrow \mathbb{R},$ define $\|f\|_{m,\mathbb{R}} := [\int_\mathbb{R} \sum_{k\leq m} |D^k f|^2 dx]^{1/2}$. Let $H_m(\mathbb{R})$ be the completion of the set $m$-times ...
2
votes
3answers
54 views

Are there any significant differences between studying functional analysis from a normed space perspective versus a metric space perspective?

Does it matter if functional analysis was introduced from a normed space versus a metric space formulation? Are all major theorems from functional analysis (such as Banach contraction mapping, Hahn ...
1
vote
0answers
26 views

Jordan normal form

Let $H$ a Hilbert space and let $T\in B(H)$ a bounded operator on H, my question is if it exist a theorem about some "decomposition" of type Jordan canonical form in a general Hilbert space, and how ...
0
votes
1answer
36 views

A general question about Cauchy operators

I want to familiarize myself more with the Cauchy operators. As soon as I say "operator" I have to specify on which space, Okay, that should be my first question: On which spaces the Cauchy operators ...
2
votes
1answer
57 views

On the space $L^0$ and $\lim_{p \to 0} \|f\|_p$

For $0 < p < \infty$, the definitions of the spaces $L^p$ are very natural. Then, we of course want $L^\infty$ and $L^0$ to be some kind of limits of $L^p$ spaces. What does the parameter $p$ ...
1
vote
1answer
24 views

Prolongement in Sobolev spaces

Let $\Omega$ be an open bounded set of $R^n$, and let $\omega$ be an open subset of $\Omega$ s.t $\overline{\omega} \subset \Omega.$ For $f\in H_0^1(\omega)$, it is known that the extension of $f$ to ...
2
votes
1answer
51 views

$\ell^p\!,$ for $p\neq2$, is not an inner product space. [duplicate]

Consider sequence spaces of the form, $$\ell^p=\Big\{x=\left(x_j\right)_{j=1}^\infty \mathrel{}\big|\mathrel{} \sum_{j=1}^\infty \left\lvert x_j \right\rvert ^p\lt\infty\Big\}$$ for $1\le ...
0
votes
1answer
21 views

Point about the theorem and proof of the inner product being a continuous function.

In moving to show that the inner product, $\langle\cdot,\cdot\rangle$ is a continuous function I have the following theorem in my notes (also on page 59 of "Linear Functional Analysis", Rynne and ...
4
votes
1answer
27 views

Complex Conjugate of Integral

I want to know that the equality $$ \overline{\int_{\mathbb R} f(x)dx} = \int_{\mathbb R} \overline{f(x)}dx $$ holds, if the both integral converges. Here $f:\mathbb R \ni x \mapsto f(x)\in \mathbb C ...
3
votes
2answers
65 views

Equivalent Norms for Intermediate Subspaces

Let $(X,\left\|\cdot\right\|)$ be a Banach space, and let $\left\{T(t) : t\geq 0\right\}$ be an equibounded strongly continuous semi-group on $X$. Define a functional ...
2
votes
1answer
54 views

Proof of the Banach–Alaoglu theorem

The Banach–Alaoglu theorem states that the closed unit ball of $B'$ (where $B'$ is the dual to a Banach space $B$ over a field) is compact in the weak* topology. I'm having trouble trying to prove the ...
5
votes
3answers
69 views

Showing $X^*$ is separable implies $X$ is separable using the Riesz lemma

If $X$ is a Banach space and $X^*$ is separable, then $X$ is separable. Here, David Mitra mentions a proof using the Riesz lemma. However, I do not fully understand it. You could also use ...
2
votes
0answers
32 views

Another equivalent characterization of Schwartz function?

Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ \sup_{x\in\mathbb{R}^n}\left||x|^k\Delta^{p}\psi(x)\right|<\infty $$ for all ...
1
vote
1answer
55 views

To be or not to be Banach? That is the question.

On the set $H^1_0((0,2))$ we put the following norms. $$\|u\|_a^2= \int_{[0,2]}(u')^2.$$ $$\|u\|_b= \|u\|_\infty.$$ $$\|u\|_c= \|u\|_{L^2}.$$ Is $H^1_0((0,2))$ Banach with any of these norms?
0
votes
1answer
29 views

How to show that $L^p$ norm is monotone increasing?

I am trying to solve the following (very standard) exercise: Let $(X,\mathcal M,\mu)$ be a measure space and $f\in L^r\cap L^\infty$ for some $1\leqslant r<\infty$. Then $f\in L^p$ for ...