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, ...

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42 views

Is there are “sphere” associated to any topological vector space?

If I have a topological vector space that is not locally compact, is it still possible to associate to it some natural "sphere" like object? For locally compact Hausdorff spaces, the my first guess ...
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36 views

Absolute continuity as a condition for $F[f^{(k)}](\lambda)=(i\lambda)^k F[f](\lambda)$

In read in Kolmogorov-Fomin's (p. 429 here) that if function $f$ is such that $f^{(k-1)}$ is absolutely continuous on any interval and if $f,...,f^{(k)}\in L_1(-\infty,\infty)$, [...] we get ...
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32 views

Relation between RKHS and space of continuous functions

Consider a Mercer Kernel $K\colon \mathcal{X}\times \mathcal{X}\to \mathbb{R}$, $\mathcal{X}$ being a compact subset of $\mathbb{R}^m$, and its (unique) associated Reproducing Kernel HIlbert Space ...
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22 views

Equivalence of condition on function

In a book I am studying it states that a condition on a function $g$ as follows: Given the function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function satisfying $$\sup_{|u| ...
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46 views

What is the Vapnik-Chervonenkis dimension of sigmoidal functions?

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...
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35 views

How to change the fundamental frequency of a sample signal?

So I am dealing with a 60Hz signal that is sampled at 1kHz. This 60Hz signal has many other harmonics (eg, 120 Hz, 180Hz..... and more). For some reason, we would like it to be 50Hz. Could we ...
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36 views

How to show this space is NOT reflexive

Consider the Banach space $X$ of null sequence whose elements are complex sequence which converges to $0$. In addition the norm is defined as $$\|(a_1, \dots, a_n)\| := \sup_n |a_n|.$$ Show this ...
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13 views

$\mathcal{G}$ equicontinuous $\implies (g_n)$ equicontinuous?

If I have a collection of continuous, real valued functions, call this $\mathcal{G}$, and say I have a sequence $(g_n)\in\mathcal{G}$. If I know that $\mathcal{G}$ is equicontinuous, can I conclude ...
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68 views

Adjoint Operator of a Compact Operator

In the proof of the fact that the ad-joint operator $T^*$ of a compact operator $T$ defined on a separable, infinite dimensional Hilbert space $\mathcal H$ is also compact, I read that "$\|P_nT - T\| ...
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48 views

Equivalent formulation for compact operators

According to Wikipedia, an operator is compact if it can be written in the form $T(u)=\sum_{n=1}^\infty \lambda_n<f_n, u> g_n$, where $\{f_n\}$ and $\{g_n\}$ are orthonormal sets and ...
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70 views

Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
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40 views

Duhamel's formulation of a pde

Let's consider the following initial value problem $$u_t=Lu+F,\,\,\,\,u(0)=u_0$$ with $L$ a spatial operator. Which are the minimal assumptions on $F$,$u_0$,$u$ to have the equivalence of the problem ...
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358 views

Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
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53 views

Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ ...
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37 views

Dual of a locally convex space

Let $X$ be a normed space. Suppose $E$ is a subset of $ X^*$ (The space of continuous linear functionals). For every $\phi\in E$, define seminorm $p_\phi: X\to [0,\infty)$ such that $p_\infty (x)= ...
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41 views

Basis for Finite Dimensional Hilbert Spaces

Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra. Here is what I know. Hilbert space ...
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38 views

Does an inequality between kernels imply an inequality between the norms of integral operators?

Assume that $g(x,y)$ and $h(x,y)$ are two positive functions such that $0<g<h$ and assume that $$T_g, T_h : L^2(B^n,R)\to L^2(B^n,R)$$ are integral operators defined by $$T_k[f](x)=\int_{B^n} ...
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143 views

Implicit function theorem to prove tangent plane to the surface

Let $\Phi$ be the regular surface at $(u_o,v_o)$ (ie., $\Phi$ is of class $C^1$ and $T_u\times T_v\ne 0$ a)Use the implicit function theorem to show that the image of $\Phi$ near $(u_o, v_o)$ is the ...
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68 views

Are there probability density functions with the following properties?

Given two distinct and continuous probability density functions on real numbers, $f_0$ and $f_1$ consider the following set of density functions: ...
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26 views

Discontinuous functionals on $L^p$

Using the axiom of choice and a Hamel basis for a normed space, one can prove the existence of everywhere defined discontinuous linear functionals. My question: Does there exist a discontinuous ...
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52 views

The spectrum of an element in a non-unital Banach algebra

Let $A$ be a commutative non-unital Banach algebra. Let $\widetilde{A}$ denote its unitisation. I am trying to understand the proof of $$ \sigma (a) = \{\tau (a) \mid \tau \in \Omega (A)\}\cup ...
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93 views

Confused about weak derivatives in Evans

I'm a bit confused about how Evans refers to derivatives at some points and if he means weak derivative. In particular on page 301 he gives the definition that if $\textbf{u} \in L^1(0,T;X)$ and ...
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145 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
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108 views

Confirm solution to chapter 2, Problem 18 in Rudin's book: principals of mathematical analysis

Is there a non-empty perfect set $E$ in $\mathbb{R}^1$ which contains no rational numbers? My effort: Yes, there is. We take $E_0 \colon = [\sqrt{2},\sqrt{3}]$. Then $E_0$ is non-empty, closed, ...
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58 views

How to prove the following isomorphism?

Let $A, B$ be two C*-algebras, $\pi:B\rightarrow A$ and $\sigma: A\rightarrow B$ be *-homomorphisms such that $\sigma\circ\pi$ is homotopic to $1_{B}$. Define a *-homomorphism $\delta: B\rightarrow ...
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29 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ ...
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47 views

Alternative proof of "Every linear mapping on a finite dimensional space is continuous”

Here is my question: Suppose that $T:X\to Y$ is linear, where $X$ and $Y$ are normed linear spaces, and $X$ is finite dimensional. Define $\|\cdot\|_\beta$ on $X$ by ...
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58 views

Relative Entropy and variation formula for $C_c$

Let $R(\mu \mid \nu ) = \int_{\mathbb R} \log \frac{d\mu}{d\nu} d\nu$ for $\mu, \nu$ probability measures over $\mathbb R$. By the varational representation formula of Donsker and Varadhan we know ...
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23 views

Show that it is a element of $(H^1(\Omega))'$

Let $\Omega$ a open regular subset of $\mathbb{R}^n$, $n\ge 1$. I want to prove that if $u\in L^2(\Omega)$, the distribution $\partial_{x_i} u $ is an element of $(H^1(\Omega))'$ (for $1\le i \le ...
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100 views

Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that ...
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63 views

Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form $$ T_0 = i \alpha \nabla + \beta,$$ where $\alpha$ and $\beta$ are Hermitian $4 \times 4$ matrices, and $T_0$ is selfadjoint ...
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57 views

Which projection, in $L_\infty$ norm or $L_2$ norm, is non-expansion?

I am just wondering which projection is non-expansion? Basically, I am wondering if $F$ is a projection operator then which norm would satisfy the following non-expansion property, where for a given ...
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35 views

Soft: Interesting examples of applications of the Itô–Nisio theorem

I'm writing up a presentation on the Itô–Nisio theorem and I'm looking for a simple (nontrivial) example showcasing it. I was thinking of a 2-dimensional simple random walk, but that seems ...
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24 views

Given that the support function $\sigma_S(x)$ is finite-valued, prove that $S$ must be bounded (without the use of Uniform Boundedness Principle)

Here is the problem: Let $S\subset \mathbb{R}^n$ and consider the support function $\sigma_S:\mathbb{R}^n\to\mathbb{R}^n\cup\{\pm\infty\}$ defined by $$\sigma_S(x)=\sup_{s\in S}(sx)$$ Prove, ...
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26 views

Bump Functions on Open Intervals

I just have a quick question about bump functions. If we're dealing with $C^{\infty}_{0}((0,\infty))$, i.e. all smooth bump functions on $(0,\infty)$, obviously any $f\in C^{\infty}_{0}((0,\infty))$ ...
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38 views

Isometric isomorphisms between normed spaces and compact hausdorff spaces

Let $X$ be a normed space. Show that there is a compact Hausdorff space $Y$ such that $X$ is isometrically isomorphic to a subspace of $C(Y)$. I think this might be proved using the Banach–Alaoglu ...
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44 views

Compositions and products on Sobolev spaces

Does anybody have a good textbook reference for someone who wants to begin studying products and compositions in Sobolev spaces, where the underlying domain is either $\mathbb{R}^n$ or an open subset ...
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44 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists ...
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63 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
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60 views

Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
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14 views

$\|\nabla f\|_p\leq C (\|\nabla \times f \|_p +\|\nabla \cdot f\|_p)$

Let $f\colon\mathbb{R^3}\to \mathbb{R^3}$ have compact support. The identity $$ -\Delta = \nabla\times\nabla \times - \nabla \nabla \cdot, $$ and two integration by parts shows that $$ \|\nabla f\|_2 ...
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26 views

Does this gradient map have a closed range?

Let $\mathbb{T}^n$ be $n$-dimensional torus. Let $H^1(\mathbb{T}^n)$ be the Sobolev space of functions in $L^2(\mathbb{T}^n)$ whose weak derivative is in $L^2(\mathbb{T}^n)$. Then the gradient map ...
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15 views

Giving a bound for |f(x) \star \phi_k(x) -f(x)|

Here is the problem: Let $\phi(x) \in S$, where $S$ is the Schwartz class, such that $\displaystyle\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \phi=1$. Also, for some $N\in\mathbb{N}$, ...
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38 views

Applying Stone Weierstrass to this isometry of $C^\ast$-algebra

I proved the following theorem but I'd like to confirm the last part of my proof. Statement: Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a ...
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35 views

When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Let $X$ be a locally convex set with the following property: (P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological ...
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36 views

exact angle between the functions $p(x)=3x-4$ and $q(x)=9x-5$ over $0\leq x\leq 1$

I was attempting a question, which gave the formula for the angle theta between two functions $f(x)$ and $g(x)$ over $a\leq x\leq b$ (note the question defined the meaning of norm and inner product as ...
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66 views

Operator for scaling a function?

Let $\mathbb{F}$ denote the set of functions of the form $f: \mathbb{R} \to \mathbb{R}$. I am interested to know whether there exists a well-known linear map $T_\alpha: \mathbb{F} \to \mathbb{F}$ ...
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63 views

On calculating spectral projections

Consider following operator from this paper; Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ...
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40 views

distance between two space

‎Given N- dimensional subspace ‎$W‎_{1}‎$ ‎and ‎‎$W‎_{2}‎‎$ ‎of a‎ ‎Hilbert ‎space‎, define the ‎$‎N$-tuple $(‎σ_{1},σ_{2},...,σ_{N}) $‎‎‎ as follows: ‎$$\sigma_{1}=\max \lbrace\langle ...
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21 views

Compact operator on non complete space - spectrum

I was trying to google it out but without succes. Is the spectrum of compact operator $f:X \rightarrow X$ at most countable in general only if $X$ is banach space ?