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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17 views

Integral of inverse function

On wikipedia and on the following mathstackexchange page, a formula for the sum of the integrals of a function and its inverse (with "corresponding" limits) is given, do you have a proper proof for ...
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39 views

geometry interpretation of uniformly rotund in every direction

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $\lim_{n\to\infty} ||x_n-y_n||=0$ whenever $x_n, y_n \in S_X$ are such that $\lim_{n\to\infty} ||x_n+y_n||=2$ and ...
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12 views

$L^p(0,T;L^p(M)) = L^p((0,T)\times M)$?

I thought it was true that $L^p(0,T;L^p(M)) = L^p((0,T)\times M)$ for $p \neq \infty$, but can't find a proof. Can someone assist me with this? Thank you
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11 views

Extend functional without changing norm

I have: $X = <\mathbb R^2, ||(x,y)|| = 2|x|+3|y|>, L = <(0,2x), x \in \mathbb R>,$ $\phi_0 \in L^* : \phi_0(0,2x) = -2x$ I need to extend $\phi_0$ on $X$ with same norm. AFAIK, I need ...
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33 views

$x_n \rightarrow 0\ (n\rightarrow \infty)$ is stable under a change of topologies

There is an example in the lecture notes I'm currently reading, in a chapter on the dual pairing of a topology, that in $E:=\ell^2$ the convergence of a sequence $(x_n)_n$ to zero in the ...
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12 views

Second derivative of Impulsive boundary value problem

I have this Impulsive problem : $$ \begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases} $$ and the associated functionnal is given by: ...
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21 views

Is this composition of functions Lebesgue measurable?

Let $Q_0 = (0,T)\times\Omega_0$ and $Q_T = \bigcup_{t \in (0,T)}\{t\}\times\Omega(t)$ where $\Omega_0$ and $\Omega(t)$ are bounded domains in $\mathbb{R}^{n+1}$. Define $F:Q_0 \to Q$ by $F(x,t) = ...
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32 views

Orthonormal set problem

A)For the First three member of $(x_0, x_1, x_2, ... ) $ with respect to $$x_{j}(t)=t^{j}$$ in $[-1,1]$ , use the inner product function below to make them orthonormal. $$\langle x,y\rangle ...
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21 views

Functional and operator associated to a problem

I have a this functional: associated to the impulsive problem : $$ \begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases} $$ ...
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12 views

Show that $\alpha_j=2iA_j$

I have a question, I don't understand that Why do we have $\alpha_j=2iA_j,\ P_j(z, \overline{z})=\text{Re}A_jz^j$ since proof of Lemma??? ======================================================= ...
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9 views

Show that $F_{ll}^{*}=\text{Re}g'_0-2l\text{Re}f_1+F_{ll}+\ldots$

I have stuck when I try to show (4.4): With $j \ge 1$, we have (4.4): \begin{align*}F_{ll}^{*}&=\text{Re}g'_0-2l\text{Re}f_1+F_{ll}+\ldots\\ ...
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18 views

bilinear form is not coercitive

Let $V=(H^1_0(\Omega))^d\times L^2(\Omega)$ with norm $\|(u,p)\|_V=\|u\|_{H^1(\Omega)} + \|p\|_{L^2(\Omega)}$. Define the bilinear form $$\begin{align*} A:& V\times V\to\mathbb R,\\ ...
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25 views

block basic equivalence in original Tsirelson space

Let $T$ denote the well-known Tsirelson space defined by the completion of $c_{00}$ under the implicit norm $\|x\|_T=\max\left\{\|x\|_\infty,\frac{1}{2}\sup\left\{\sum_{i=1}^n\|E_ix\|_T:n\leq ...
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18 views

measure space and uniform boundedness principle

If $X$ is a locally compact and $\{\mu_n\}$ is a sequence in $M(X)$, then $L(\mu_n)\to 0$ for every $L\in M(X)^*$ iff $\sup_n||\mu_n||<\infty$ and $\mu_n(E)\to 0$ for every Borel set $E$. I've ...
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31 views

Integration by parts with outer product

Let $$K:=\{m_{ij}\in L^2(\omega)|i,j\in\{1,2\}\}$$ be equipped with the natural inner product $$(x,y)_K:=\sum_{i,j=1}^{2}(x_{ij},y_{ij})_{L^2(\omega)}.$$ If the functions are regular enough, is it ...
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42 views

How to make sense of the Fourier transform of this distribution

I want to compute the Fourier transform of this distribution: $$D(f)=\int_{\mathbb{R}} f(t,t^2) \frac{dt}{t}$$ ($f$ a Schwartz function on $\mathbb{R}^2$, the integral interpreted with a Cauchy ...
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17 views

measure space and dual of continuous functions

I know that $M(X)$ (the space of complex-valued measures ) is isometrically isomorphic to $C_0(X)^*$. thus for every measure $\mu$ there is a bounded linear functional $F$ such that $F(g)=\int g \, ...
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27 views

Using the principle of uniform boundedness

From wikipedia we have Theorem (Uniform Boundedness Principle): Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. If ...
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40 views

Dilation and translation of the Dirac Delta distribution

Given $\delta(ax) = \frac{1}{|a|}\delta(x)$ and $\delta(ax-b) = \frac{1}{|a|}\delta(x-b/a)$ , it it true also that $\delta(a(x-b)) = \frac{1}{|a|}\delta(x-b)$ ?
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12 views

Prove that the closure of the Choquet boundary is contained in every closed boundary

I am reading the proof of the above in Andrew Browder's 'Introduction to Function Algebras'. Here is what he says: Let $Y$ be a closed boundary of a function algebra $A$, a subset of $C(X)$ for $X$ ...
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24 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
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38 views

Why Laplace transfrom uses Exponential function

I probably don't have enough background to post this question, but I am very curious about it. The way I think about the Laplace transform now, the Laplace transform multiplies a given signal by ...
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38 views

Impulsive Boundary value problems

I have this paper They consider this impulsive problem i dont understand this : Proof. First, suppose that $x\in E\cap C^2[J',R]$ is a solution of problem $(1.5)$. It is easy to see by ...
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24 views

Saturated Monotone and Increasing Mappings

Let $A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a monotone mapping, i.e., $$ \left( A(x) - A(y) \right)^\top \left( x-y\right) \geq 0 $$ for all $x,y \in \mathbb{R}^n$. Let $B : \mathbb{R}^n ...
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32 views

Separability of a certain space of continuous functions

Let $I$ be a separable, locally compact Hausdorff space, and let $V$ be a separable, locally convex, complete topological vector space. Consider the function space $C(I, V)$ with the compact-open ...
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25 views

Question about meaning of evolution problem.

Consider the following "evolution problem" $f(t) - u_t(t) \in \partial \psi(u(t))$ $u(0) = u_0$ Where $f:[0,T] \rightarrow H$ $ u:[0,T] \rightarrow H$ $ \psi:H \rightarrow (-\infty,\infty]$ is ...
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26 views

Prove operator convergence

I have to proove that given $X$ a normed space and $Y$ a Banach space, if the sequence of bounden linear operators from $X$ to $Y$ $\{A_n\} \rightarrow A$ and the sequence $\{x_n\} \rightarrow x$ then ...
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27 views

What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
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26 views

Dual of a Sobolev space (ex)

Consider $f(x_1,x_2)=\chi_{B_1(0,0)}(x_1,x_2)$. 1) Is $\nabla f\in (H^1_0(\mathbb{R}^2;\mathbb{R}^2))^*$? 2) $\langle \nabla f , u \rangle= ?$ with $u\in H_0^1(\mathbb{R^2})$? For the first ...
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25 views

Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
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27 views

Relation between Gateaux derivative and rotund

I want to show that if a Banach space $(X, ||.||)$ reflexive and $||.||$ is an equivalent Gateaux differentiable norm on $X$.Then its dual norm $||.||^*$is rotund. Proof:Let $x\in S_X$ and $x^*_n, ...
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70 views

k-times differentiable functions on [0,1]

Is $C^k[0,1]$ (the set of all k-times differentiable function, not necessarily continuously) complete with respect to the norm $\|f\|_\infty + \|f'\|_\infty +\cdots+\|f^{(k)}\|_\infty$? I know the ...
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40 views

$\ell^p\subset\ell^q$ if $1<p<q<\infty$

I need a reference states that $\ell^p\subset\ell^q$ if $1\leq p<q<\infty$. I could find the result on wikipedia and some homework sets but I need to cite this in a paper I am writing.
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29 views

How to prove that a sequence in a normed space tend to a limit outside of the space.

We are given a normed space $(V, ||.||)$ and a sequence $(X_n)_{n=1}^{\infty}$ in it. Suppose I can show that $X_n \to L$ under the norm $||.||$. That is, $||X_n - L|| \to 0$ as $n \to \infty$. Now, ...
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29 views

If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
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42 views

$C^1[0,1]$ is not complete with respect to sup norm

I think the right sequence is $f_n(t)=|t-\frac{1}{2}|^{(n+1)/n}$ but I can't manage to prove it's Cauchy... (when I look at graph of $f_n$s it seems obvious, but I need it formal way.
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17 views

If $f$ is a bounded linear functional and $ \|f \| \neq 0 $ then $f(h) \neq o(h)$

I am working on an example from D. H. Griffel's Applied functional Analysis (p. 309) book. But I just can't seem to understand the justification for the following example in the book?. Claim: If ...
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52 views

Prove that $C^1[0,1]$ is complete

I have to prove that $C^1[0,1]$ is complete with respect to the norm $\|f\|=|f(0)|+\int_0^1|f'(t)|dt$ is complete. My attempt: Let $f_n(t)$ be a Cauchy sequence from $C^1[0,1]$. We have: ...
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11 views

Rearranging 2 discriminant function to solve for 1 parameter (to derive a decision boundary)

I have a task where I want to classify patterns from 2 classes where the samples are drawn from a bivariate Gaussian distribution. I use the 2 discriminant functions ($g_1$ and $g_2$) to classify the ...
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26 views

injection in H^{-1}

let $\Omega$ an open on $\mathbb{R}^n$. if $f \in H^{-1}(\Omega)$ and $g \in L^1(\Omega)$. Who is the Sobolev space $V$ who can contains $f-g$ such as $V$ is injected on $H^{-1}(\Omega)$? Thanks for ...
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21 views

Set of limit points of Riemann Integrable functions

I've looked around for answers to this question. It seems like perhaps I don't have enough knowledge of functional analysis to figure out the answer (or even understand the answer), but I'm intrigued. ...
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28 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
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41 views

Is Arzela-Ascoli with equicontiuous or uniformly equicontinuous

I am still working on this proof of Arzela-Ascoli but now I noticed that in my statement of the theorem I used ''equicontinuous'' to mean ''uniformly equicontinuous''. At least in the direction I ...
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26 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
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77 views

Continuous compactly supported real valued functions on a locally compact and $\sigma$ compact space is separable

I already know that if $X$ is a compact metric space then the space of continuous real valued functions $C(X \to \mathbb R)$ are separable. What I'm trying to prove that if $X$ is a locally compact ...
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56 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
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0answers
26 views

Use Fatou theorem to prove convergence

let $u_n$ an sequence uniformly bounded in $H^1_0(\Omega)$, then, $u_n$ converge weakly to $u$ in $H^1_0$, and strongly in $L^2(\Omega)$ and a.e $x \in \Omega$. Let $g(x,u)$ an Carathedory function ...
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31 views

Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
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0answers
10 views

How does invariance of $q$ wrt $\lambda$ for a stationary functional, restrict the function?

Suppose I have the following functional: $$S(q) = \int_{b}^{a}L(t, q(t), q'(t))dt$$ and $q(t) = x(t) + \lambda$, where $\lambda$ is a constant independent of t. If $S(q)$ is stationary for a ...
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23 views

What is the contour lines of $\frac{2x+y}{2x-y}$?

What is the contour lines of $$f(x)=\frac{2x+y}{2x-y}$$? I need help to describe them... Id like to get help... Thank you!