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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Differentation Operator

having trouble completing the proof for this question Let $D:\mathbb{R}[X] \to \mathbb{R}[X]$ be the differentiation operator $D(f(X))=f'(X) .$ Prove that $e^{tD}(f(X)) = f(X+t)$ for $t \in ...
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1answer
130 views

Norm of a Kernel Operator

So, I was practicing some problems and considered the space $X = C[a,b]$ with the $L_{1}$-norm. I consider the operator $$Tf(x) = \int_a^b k(x,y)f(y)\,dy$$, where $k(x,y)$ is continuous in both of ...
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1answer
130 views

Who can help me estimate the operator norm of this integral operator?

The operator $T_F(x)$ depending on the function $F\in L^1\cap L^2$ and the real number $x$ is formally defined as follows: $$ T_F(x)\psi(y)=\int_0^{\infty}\psi(t)F(x+t+y)dt $$ Now my question is: ...
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space of schwartz, problem [duplicate]

my question is: Let $f\in S(\mathbb{R})$, with $f(0)=0$, then there exists $g\in S(\mathbb{R})$ such that: $$ f(x)=xg(x)\;\text{ for all }\;x\in \mathbb{R}.$$ I need to prove this.
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911 views

Proof of Sobolev Inequality Theroem

I have a short question about the proof of Theorem 2 below. I have included Theorem 1's statement since it is used in the proof of Theorem 2. Definition: If $1 \leq p < n$, the Sobolev Conjugate ...
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1answer
145 views

Question related to the spectrum of a bounded operator

If $A$ is a bounded linear operator on a Banach space $X$ and $\lambda\in \sigma(A)$, is it true that for all $\epsilon>0$, there is $ x\in X$ and $||x||=1$ such that $$ ||(A-\lambda I) x|| ...
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2answers
335 views

Uncountable orthonormal system in Hilbert spaces

I need an example of a Hilbert space in which the following does not hold for all $x$: $$ x=\sum_k^{\infty} \langle x,u_k \rangle u_k. $$ That is, there are elements that are not expressible as ...
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1answer
70 views

Series constructed from a cauchy sequence

Given a cauchy-sequence $\{x_i\}_{i\in \mathbb{N}}$ in a normed space $X.$ I need to construct a series that converges in $\mathbb{R}$ with $\{y_i\}_{i\in \mathbb{N}}$ a sequence in $X$: ...
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1answer
101 views

Bounded linear mappings in Hilbert space preserve orthogonality?

My question is the title of this thread! Assume we have a bounded, linear mapping $A:H\to H$ where $(H,\langle\cdot,\cdot\rangle)$ is a Hilbert space, and two non-zero elements that are orthogonal, ...
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1answer
112 views

Chebyshev polynomials with non-negative constants

Please let help me solve the following problem that I encountered while engaging in my research. I'm dealing with a class of functions, in which each function has a unique series representation of ...
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1answer
70 views

Do $L^P$ functions form a metric space?

I have a general question about $L^P$ functions. I have heard that $L^P$ functions form a vector space. My question is can we make them form a metric space too? And what is/are the possible metric/s ...
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1answer
78 views

Convergence and Constant sequence?

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0 x_1 x_2...$ given by $x_n = g(x_{n-1})$. converges ...
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1answer
143 views

$C_0^\infty(0,T)\cdot V$ dense in the Bochner space $L^2(0,T;V)$

Let $V$ be a Banach space and $(0,T)$ a time interval. Consider the space $C_0^\infty(0,T)$ of infinitely often differentiable functions with values in $\mathbb R$ and compact support in $(0,T)$ and ...
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1answer
176 views

Has a probability density function a weak derivative

Assume I have a probability density function $\rho$ on $R$. (e.g $\rho \geq 0$ $\int \rho dx =1 $ $\rho \in L^1(R)$ ...). So $\rho$ is the density wrt the lebesgue measure. Now I try to understand if ...
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1answer
137 views

Weighted $\ell^2$ space is Hilbert

This is my exam's question that I could not solve it. Please help me to undrestand how to solve it: let $\{w_n\}$ is a positive real numbers sequence, and let $$\ell^2(w):=\left\{\{x_n\}:x_n \in R ...
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1answer
127 views

closed bounded equivalent to compactness?

In preparation for an exam I am struggling with the following problem. We let $A:=\{x=(x_{n})_{n}\in \ell^{2}| \phantom{x} \|x\|\leq 1\}$ and consider $d:A\times A\rightarrow \mathbb{R}_{+}$ ...
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1answer
50 views

How to prove, that the ordering on positive bounded operators agrees with ordering of their ranges?

Hypothesis: Assume, that $A$ and $B$ are positive bounded operators (on some Hilbert space $H$) and $A\geq B \geq 0$. Then ${\rm range}(A) \supset {\rm range}(B)$. The textbook "$C^*$-algebras by ...
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1answer
52 views

About a Lipschitz condition.

In his book Differential Manifolds, Lang states the following: If $f : U \subseteq E \rightarrow E~$ is a $C^1$ map, where $E$ is a Banach space and $U$ is an open set, then it follows at once ...
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1answer
129 views

Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
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122 views

Dimension of space of continuous functions

Please can you help me in this exercice. Prove that the normed space of continuous functions $f: [0,t] \to \mathbb{R}$ with the fundamental norm $||\cdot||_2$ is infinite dimensional. ...
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1answer
98 views

A variance-mixture model

So I've tried to make a probability distribution which has a tunable degree of kurtosis and which becomes Gaussian if the control-parameter goes to 0. Now Levy-distributions are out of the question, ...
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1answer
197 views

Why are the Differential- and multiplication mapping on $C^{\infty}(\Omega)$ continuous?

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
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1answer
159 views

Density of smooth functions in fractional Sobolev space

I am reading a paper on the analysis of numerical methods, and am confused about a statement made. I am working in fractional Hilbert spaces, but I don't think that this has much bearing on the answer ...
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1answer
120 views

An quasi-nilpotent operator restricted to a subspace is a nilpotent?

I am reading a paper about operator theory, there is a proposition I could not understand. Let $T\in L(X)$ be a quasi-nilpotent operator and $X_{1}$ be a non-zero finite-dimensional subspace of X, ...
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1answer
93 views

Why is any norm-bounded family on a reflexive Banach space relatively weakly compact?

Why is any norm-bounded family $T \subseteq L(X)$ on a reflexive Banach space $X$ relatively weakly compact?
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1answer
157 views

Contradiction achieved with the Pettis Measurability Theorem?

$\bf{\text{(Pettis Measurability Theorem)}}$ Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure. The following are equivalent for $f:\Omega\to X$. (i) $f$ is $\mu$-measurable. (ii) $f$ is ...
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1answer
85 views

What is the standard (?) operator norm usually used in functional analysis?

I am studying introduction to functional analysis, in my lecture notes I have seen that a norm on functions is used in some proofs. For example I have seen the following: We note that for every ...
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1answer
333 views

Is there a non-reflexive Banach space which is strictly convex?

I just come up with the fact that a space being strictly convex, does not implies it is reflexive (at least I never saw a proof of it). How can one construct a example of a non-reflexive Banach ...
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1answer
55 views

Help showing $\phi _k$ is a bounded linear functional

Let $V$ be the space of continuous functions on the interval $[-\pi , \pi]$ with the $L^2$ norm $$\lVert f\rVert_2=\left(\int_{-\pi}^\pi |f(t)|^2\mathrm dt)\right)^\frac{1}{2}$$ For $f$ in $V$, define ...
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1answer
403 views

Adjoint of a multiplication operator

Let $B$ be the Banach space of continuous functions vanishing at infinity and defined on a locally compact Hausdorff space $X$. Given a continuous and bounded function $g$ on $X$, let $T$ be the ...
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1answer
72 views

Continuity of an additive operator on Banach spaces

If operator $A: X \to Y$ satisfies $A(u+v)=Au+Av$ for all $u,v \in X$ and $A$ is continuous at one point of $X$ then $A$ is continous. Here $X$ and $Y$ denote Banach space. I showed if $A$ is ...
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1answer
53 views

$f\in C([0,T];C^1(\Omega))$ implies $f \in C([0,T];W^{1,p}(\Omega))$?

Let $f \in C([0,T];C^1(\Omega))$ where $\Omega$ is compact. Am I correct that this implies $f \in C([0,T];W^{1,p}(\Omega))$ for all $p > 1$?? Because we have $L^\infty$ estimates on $f(t)$.
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1answer
299 views

Compactness of Hilbert-Schmidt Operator

I'm trying to show that a certain Hilbert-Schmidt operator is compact following some exercises in Rudin's Functional Analysis (exercise 15 on page 112): If $X, \mu$ is a finite measure space and $K ...
2
votes
1answer
63 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
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1answer
57 views

Defining $W^{k,p}(M)$ for non-integers $k$ and $p$ and manifold $M$

For $k$ and $p$ not necessarily integer, and on a smooth manifold $M$, how to define the Sobolev space $W^{k,p}(M)$? I've only seen definitions for $p=2$.
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1answer
151 views

On Some Locally Convex Topologies of a Vector Space

Suppose that $(X,\tau)$ is already a locally convex TVS. Let us denote by $X'$, the space of all $\tau$-continuous linear functionals on $X$, the topological dual of $X$. For each $f\in X'$, define ...
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1answer
49 views

A distribution problem

Let $\phi\in L^2(\mathbb{R}^3)$. Since $|\phi|^2\in L^1$, it has a distributional derivative. At least formally, $$ \nabla |\phi|^2 = \phi \nabla\overline{\phi} + \overline{\phi}\nabla\phi. $$ Is ...
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1answer
255 views

Functional analysis and Quantum Mechanics

I am presently doing a course on functional analysis. I have done courses on quantum mechanics before. I see that many functional analysis books have an ending chapter on quantum mechanics. So are ...
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2answers
269 views

Stone Weierstrass Overkill in the Measurable Setting?

If $\mu$ is Lebesgue measure on the Borel sigma algebra $\mathcal{B}$ of $[0,1]$. Establishing that the linear span in $L^{2}([0,1]\times[0,1],\mathcal{B}\otimes\mathcal{B},d(\mu \times \mu))$ of the ...
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1answer
115 views

Need help with one basic result of linear operators on Banach spaces

$X$ and $Y$ are Banach spaces and suppose that $T$ be a closed subspace of $X$. $A:X \to Y$, $B:Y\to X$ and $X_{0}:Y\to X$ are linear bounded operators. It is given that the operator $BA$ is ...
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1answer
390 views

Can I use inverse inequality for an infinite space, $H^1$?

Let $u \in H^1(\Omega)$ and $Q_0 u$ is a $L^2$ projection of $u$ to a polynomial finite space $P_k(T)$ where $T \in \mathcal{T}_h$ is a finite element and $\mathcal{T}_h$ is a set of all the elements. ...
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1answer
58 views

Different notions of differentiability

The following is somewhat unclear to me. Let $X$, $Y$ be locally convex vector spaces, let $f: X \supseteq U \longrightarrow Y$ be a (nonlinear) continuous map. Then one can say that $f$ is $C^1$ if ...
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1answer
787 views

Invertibility of a linear operator on a Hilbert space.

Let $H$ be an infinite dimensional Hilbert space over $\mathbb C$, $T$ be a continuous linear operator of $H$, $r(T)=\sup_{||x||=1}|(Tx|x)|$ be the numerical radius of $T$, and $z\in \mathbb C$, such ...
2
votes
1answer
51 views

Restriction to $\mathbb{R}^{d-1}$ as an operator on $L^2(\mathbb{R}^d)$

Identify $\mathbb{R}^{d-1}$ with $\mathbb{R}^{d-1}\times \{0\}\subseteq \mathbb{R}^d$. Is there a bounded operator $T: L^2(\mathbb{R}^d)\rightarrow L^2(\mathbb{R}^{d-1})$ such that $T(\phi)=\phi ...
2
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1answer
169 views

Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
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1answer
64 views

If $X$ is a normed space and $Y \subset X$, show $\max\limits_{\substack{f \in X^*,\\ \|f\|\leq 1,\,f|_{Y}=0\;}} |f(x)|=\inf\limits_{y \in Y}|x-y|$

Let $Y \subset X$ a subspace of normed space $X$. Show that $$\displaystyle \max_{f \in X^*, \ ||f||\leq 1, \ f|_{Y}=0} |f(x)|=\inf_{y \in Y}|x-y|.$$
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1answer
164 views

approximation property

In I. Namioka and R. R. Phelps's your paper "Tensor products of compact convex sets" Pacific Journal of Mathematics, Vol. 31, No. 2, 1969), they gave the following definition of approximation ...
2
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1answer
43 views

What will happen when we move the time integration out of space norm?

Given $f(\mathbf{x},t)\in L^2\big((t_1,t_2);\mathbf{L}^2(\Omega)\big)$, how to prove the following inequality? $$ \Bigg\|\int_{t_1}^{t_2}f(\mathbf{x},t)dt\Bigg\|_{\mathbf{L}^2(\Omega)} \le ...
2
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1answer
262 views

About functions of bounded variation

I got the following the following idea in one of the articles that I'm reading. It goes this way. Let $X$ be a Hausdorff topological vector space and let $\mathcal{D}$ be the family of all divisions ...
2
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1answer
247 views

Simplification of integral with division between summations

Considering that $$\sum_{j = 0}^{\infty} \int f_j(x) < \infty$$ and $$\sum_{j = 0}^{\infty} \int g_j(x) < \infty$$, $\forall x \in \mathrm{R} : f(x) \gt 0, g(x) \gt 0$. How can I simplify the ...