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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78 views

Inclusion in $L_p$ space

I have been wondering how to prove the following statement, and would greatly appreciate your help: If $f$ is a bounded function on $E$ that belongs to $L_{p_1}(E)$, then it belongs to $L_{p_2}(E)$ ...
4
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0answers
82 views

Is the result still true if we drop completeness? [duplicate]

I know how to prove the following exercise ( from Folland) : If $X$, $Y$ are Banach spaces. $T:X\rightarrow Y$ is a linear map such that $f\circ T\in\operatorname{dual}(X)$ whenever $f\in ...
3
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2answers
400 views

Proving that the Fourier Basis is complete for C(R/$2*\pi$ , C) with $L^2$ norm

Let $H$ be the inner product space = $\{f: \mathbb{R} \to \mathbb{C} \mid f \text{ is continuous and has period }2 \pi\}$ where the inner product is: $$\langle f,g \rangle = ...
2
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1answer
727 views

Norm equivalence Sobolev space

I have this problem: Let $k>0$ (integer) and $1 \leq p < \infty$. Show that the norms $$ ||u||_{W^{k,p}(U)} = \bigg( \sum_{|\alpha|\leq k}||D^{\alpha}u||_{L^{p}(U)}^{p}\bigg)^{\frac{1}{p}} $$ ...
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79 views

Local integrability of a Cauchy transform in the plane

Let $\mu$ be a Radon measure in the plane, typically with support included in a small neighborhood of the origin. Let $h(z)=\int \frac{d\mu(y)}{z-y}$. I am wondering when it can be said that ...
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45 views

a functional space for the study of exponential sum

I would like to study the exponential sum in an appropriate functional space. In particular: $f(x): \mathbb{R} \to \mathbb{R} $ $f(x) = \chi_K \sum_{i=1}^{M} R_i \exp{(s_i x)}$ where $R_i, s_i \in ...
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1answer
102 views

Is this gradient an isomorphism on its range?

Consider the gradient (in the weak sense) as an operator $\nabla \colon H^1(\Omega)/\mathbb{R} \to [L^2(\Omega)]^d$, where $\Omega \subset \mathbb R^d$ is a domain with a smooth boundary and ...
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2answers
103 views

Aronszajn's criterion for euclidean space, again

Referring to the paper Aronszajn's Criterion for Euclidean Space by R.D. Arthan: could someone explain or simplify lemma 3 in all if you are happy to do that? I am in desperate need for that. My ...
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1answer
78 views

Density of Sobolev spaces

Is the Sobolev space $H^s(\mathbb{R}^3)$ dense in $L^2(\mathbb{R}^3)$ for $s>0$ ?
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1answer
231 views

Different Types of Continuity in Reflexive Banach Space

Let $X$ be a reflexive Banach space with dual $X^*$. Let $K\subset X$ be a nonempty closed convex set. The mapping $F: K\rightarrow X^*$ is said to be: weakly continuous if $F$ is continuous w.r.t. ...
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101 views

$\int fg = \lim_{n \rightarrow \infty} \int f_n g.$

From an old notebook of mine, I saw this unproved exercise in class: Let $f_n$ be a uniformly bounded sequence in $L^p$ ($1<p<\infty$) such that $f_n \rightarrow f$ a.e. in $L^p$. Then for ...
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50 views

Definition of continuity for operators $L^2_{0} \to L^2_{loc}$

Let $L^{2}_{loc}(\Omega)$ be a space of measurable functions on domain $\Omega$ in $\mathbb{R}^n$ with topology induced by a family of seminorms $$ p_{\alpha}(u) = \| ...
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2answers
64 views

There does not exist $c \ge 0$ such that $\|f\|_{\max} \le c\|f\|_1$

Let $f \in C[a,b]$ and let $\|f\|_1$ be the $\mathcal{L}^{1}$-norm and $\|f\|_{\max} = \max_{x \in [a,b]}|f|$. They are both norms on the given vector space. I want to prove that $\not \exists c \ge ...
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4answers
622 views

Understanding Complete Metric Spaces and Cauchy Sequences

From my own definition, I have concluded that a complete metric space is a set and a metric where the set consists of no holes in it. Book definitions describe that "A complete metric space is a ...
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0answers
80 views

necessary and sufficient conditions for a subset to be the graph of a linear operator

Let $X$ and $Y$ be two linear vector spaces. Find necessary and sufficient conditions for a subset $G\subset X\times Y$ to be the graph of a linear operator from $X$ into $Y$. The definition for the ...
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59 views

Dense subset of the conjugate space

The question is: Let $X$ be a normed linear space and let $B$ be a dense subset of $X^*$(the conjugate space of $X$). If a sequence $\{x_n\}$ in $X$ is bounded, and if $\lim_n x^*(x_n)$ exists for ...
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1answer
52 views

functional calculus $\theta(1/f)=\theta(f)^{-1}$

If $T\in B(H)$ is normal,and if $f\in C(S_p(T))$ is never zero,how to prove that the functional calculus $\theta$ for T satisfies $$\theta(1/f)=\theta(f)^{-1}$$
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1answer
109 views

the spectral radius in a unital Banach algebra

If x,y are commuting elements in a unital Banach algebra,how to prove that $r(x+y)\leq r(x)+r(y)$and $r(xy)\leq r(x)r(y)$,where r is the spectral radius
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1answer
260 views

natural embedding of normed linear space is an isometry

To review for an exam, I'm trying to write up a short proof of the following: Let $J: X \rightarrow X^{**}$ be the natural embedding of the normed linear space $X$ into its bidual $X^{**}$, ...
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1answer
52 views

Are there $L^2$ functions on the boundary of the disk that are not in the image of the given extension?

Consider the Sobolev space $W^1$ that is the closure of $\mathcal{C}^{\infty}(\bar{D_1})$ with respect to the norm $$|\phi|_1^2=|\phi|^2_{L^2(\bar{D_1)}}+|\nabla \phi|^2_{L^2(\bar{D_1)}}.$$ I have ...
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1answer
43 views

Induced linear operator on the dual space

I have a general question. Suppose you have a bounded linear operator $T: (L^p)^* \to (L^{q})^*$. Is there always a bounded linear operator $T': L^q \to L^p$ naturally induced by $T$? If no, are there ...
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1answer
209 views

Weak holomorphicity implies smooth and holomorphic.

This is an extension of a previously asked question: A function $f\in L^2(D)$ is weakly holomorphic if, for every $\phi\in \mathcal{C}^{\infty}_c(D)$, $$\int_D f\partial_{\bar{z}}\phi = 0.$$ I'm ...
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1answer
495 views

Isometric isomorphism

In the case that $L:B_1 \rightarrow B_2 $ is a linear mapping of Banach spaces and $L$ is a isometric isomorphism (bijection and $||Lx||_{B_1} = ||x||_{B_2} $) can I say that $L\overline{L}= 1 $ is ...
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1answer
101 views

Sobolev spaces doubt

Can somebody help me with this doubt? Let $\Omega$ an open set and $A$ be any finite subset of points of $\Omega.$ Is it true the following inequality? $\vert v(a) \vert \leq C \| v \|_p ...
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2answers
709 views

Dual space of $H^1$

It holds that $W^{1,2}=H^1 \subset L^2 \subset H^{-1}$. This is clear since for every $v \in H^1(U)$ $u \rightarrow (u,v)_{H^1}$ is an element of $H^{-1}$. Moreover for every $v \in L^2(U)$ $u ...
3
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1answer
117 views

Banach-Steinhaus variant

Let $T_n$ be a sequence of continuous linear operators from a Banach space $X$ to a normed linear space $Y$. Now, for all $x \in X$, $\lim_{n \rightarrow \infty} T_n(x)$ exists in $Y$. Define ...
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1answer
413 views

Uniform boundedness principle proof

Let $\lbrace T_i \rbrace$ be a family of continuous linear operators from Banach space $X$ to normed linear space $Y$. If for all $x \in X$, there is an $M_x \geq 0$ for which $||T_i(x)|| \leq ...
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1k views

Are vague convergence and weak convergence of measures both weak* convergence?

For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due ...
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793 views

Tensor product of operators

We know that if $T_1$ is a linear bounded operator on a Hilbert space $H_1$ and $T_2$ is a linear bounded operator on a Hilbert space $H_2$ there exists a unique linear bounded operator $T$ on $H_1 ...
2
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1answer
90 views

Cohomology with Coefficients in the sheaf of distributions

It just occurred to me that one could form the sheaf of distributions $F$ on any manifold where for an open set $U$ we have $F(U)$ is the algebra of distributions on $U.$ What does cohomology with ...
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282 views

Relation between weak topology and topology of weak convergence?

From Wikipedia, the weak topology on a topological vector space $X$ is the initial topology with respect to its continuous dual $X^*$. In other words, it is the coarsest topology (the topology with ...
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484 views

closed graph theorem proof

I'm trying to prove this: Let $T:X \rightarrow Y$ be a linear operator between Banach spaces. Then $T$ is continuous if $T$ is closed. Here's my take: $T$ being closed is equivalent to the ...
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1answer
80 views

Need an example when the space of bounded linear maps is not complete [duplicate]

I know that if Y is complete then the space of bounded linear maps from X to Y is complete. What happens if X is complete but Y is not complete ? Can you please give me an example where the space of ...
0
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1answer
61 views

Weak Holomorphicity: Notation clarification.

A function $f\in L^2(D)$ is weakly holomorphic if, for every $\phi\in \mathcal{C}^{\infty}_c(D)$, $$\int_D f\partial_{\bar{z}}\phi = 0.$$ I'm trying to show that each such $f$ is smooth on the ...
3
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1answer
280 views

Retarded Functional Differential Equation

I would like some help understanding the proof of lemma 6.1 given here, for the case of an autonomous Retarded Functional Differential Equation (RFDE). The problem for the autonomous case (In the ...
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1answer
186 views

Orthonormal basis in Hilbert spaces

I have a general question but I'm going got ask it in a very restrictive setup. It is known that an equivalent condition for a system $\left\{e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ being an ONB ...
2
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1answer
191 views

Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$?

Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$? More precisely, I want to prove that THEOREM. A sequence $\{f_n\}$ is convergent in $C^k(\bar\Omega)$ (or some more ...
2
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3answers
368 views

Relation between $T=0$ and $(Tx,x)=0$

Let $X$ be a vector space and (.,.) be an inner product on $X$ also if we have a linear operator $T:X\rightarrow X$, then in both cases real and complex for inner product what is the relation between ...
4
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1answer
80 views

Strong convergence of multiplication operator

I am looking for a necessary and sufficient condition for a sequence of multiplication operators $T^{(k)}$ to converge to zero strongly. (i.e. $\forall x \in \mathcal{H} \quad ||T^{(k)}x - 0|| \to 0$ ...
0
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1answer
50 views

If $A$ is invertible, so is $A^*A$

Let $A \in L(H)$, for a Hilbert space $H$. If $A$ is invertible, why is $A^*A$ invertible, too?
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179 views

What tools are used to show a type of convergence is or is not topologizable?

There are many types of convergence. For example, in measure theory and probability theory, there are many types of convergence of measurable mappings (random variables). in measure theory and ...
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265 views

Can a non-zero vector have zero image under every linear functional?

Let $X$ be an infinite-dimensional vector space, and let $x_0$ be an element of $X$ such that $f(x_0)=0$ for every linear functional $f$ defined on $X$. Then can we prove that $x_0$ is the zero vector ...
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357 views

How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?

Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
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3answers
649 views

How to find the norm of this bounded linear functional?

Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
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67 views

Is there a canonical probability measure on smooth curves?

For continuous curves, we have Brownian motion giving the most natural probability measure. However, the sample paths of Brownian motion are almost surely terribly behaved (not of bounded variation, ...
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40 views

Difficult Series to analyse

I have the following series, $$ \sum_{t=N}^\infty F(t,N),$$ $$ F(t,N) = Max \, \{ \, \, \,1 - A(t) \, \,e^{-\frac{ N \,b}{ t} }\, , \, \, \, \, 0\, \, \, \} $$ where $A(t)$ cannot grow faster ...
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1answer
27 views

What is $| g(t,x) |$ for multidimensional $g$?

I'm reading a book on ODE, and find $|\cdot|$ is confusing. It says: Consider a function $g:\Omega \rightarrow \mathbb{R}^n$. For every compact $K\subset \Omega$, there exist constants $C$ and $L$ ...
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1answer
103 views

Which one is the right definition of eigenvalues for a differential operator?

This question might be trivial, but I have problems understanding the definition of eigenvalues for the Laplacian \begin{equation} \Delta : C^2(U) \to C(U). \end{equation} on some open, bounded domain ...
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73 views

Set-valued map (measurability)

I have this exercice and i want to know how to solve it : 1)- Let $X,Y$ two separable metric spaces ,let $(\Omega, \mathcal{A})$ be a measurable space ,and $f: \Omega \rightarrow X$ a measurable ...
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1answer
131 views

Infimum of a Hilbert space inner product

This is exercise 5.11 in Brezis's Functional Analysis, Sobolev Spaces, and PDEs. Let $H$ be a Hilbert space, and let $M \subset H$ be a nonzero closed linear subspace. Let $f \in H$, $f \notin ...