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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98 views

Norm of functional via dual space?

Suppose $f \in X^*$, where $X$ is a Banach. Is there such a result: $$\lVert f \rVert_{X^*} = \sup_{x \in X, \lVert x \rVert_X = 1} |l(f,x)|$$ where $l(f, \cdot):X \to \mathbb{R}$ is bounded and ...
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85 views

Need help with Gronwall lemma consequence

suppose we have an equation $$\frac{d}{dt}\xi(t) + a(t) = 0.$$ Integrating $$\xi(T) + \int_0^T a(t) = \xi(0)$$ and using a bound on $a$, we have $$\xi(T) \leq \xi(0) + \beta\int_0^T\xi(t)$$ Applying ...
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110 views

Why does the norm topology makes all linear functions continuous?

Assuming we have a normed vector space V (assume infinite dimensional, as trivial if finite dimensional), then why does the norm topology make all linear functionals on V continuos? I can't see how ...
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48 views

Optimal distribution with moment conditions

Basically, I want to find a probability distribution which maximizes a convex objective function and satisfies two moment constraints. For given $\bar x$, $m_{n-1}$, $m_n$ $$ \max_{f(x)} ...
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79 views

The deficiency indices of symmetric operators

Given any pair of nonnegetive integer $(a,b)$, can you find an (unbounded) symmetric operator $T$ with the deficiency indices $(a,b)$? I guess the answer is yes, but how to do it?
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550 views

Rademacher function and weak convergence

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
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162 views

Proof of compactness of bounded linear operator

Define $T: l^2 \to l^2$ by $Tx = y =(\eta_j)$, where $x = (\xi_j)$ and $$ \eta_j = \sum_{k=1}^{\infty} \alpha_{jk}\xi_k, \quad \quad \sum_{j=1}^{\infty} \sum_{k=1}^{\infty} |\alpha_{jk}|^2 < ...
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44 views

How can I prove $\mathcal S$ is dense in $W^{s,2}$?

Let $\mathcal S (\Bbb R^n)$ be the Schwartz class and $W^{s,2}(\Bbb R^n)$ be the Sobolev space($s=0,1,\cdots$). In fact I know that $C_c^\infty(\Bbb R^n)$ is dense in $W^{s,2} (\Bbb R^n)$ and ...
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217 views

Integral operators with operator valued kernels

This is the definition for integral operators I know: Let $\Omega \subset \mathbb{R}^n$ and $D \subset \mathbb{R}^n$. Let $K : \Omega \times D \to \mathbb{C}$ be measurable. A linear operator $T: ...
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69 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
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33 views

Multiplicationoperator self-adjoint on given domain

I have a question. Given the Hilbertspace $H=l^2(\Bbb{N})$ with multiplicationoperator $T_f$, with $f:\Bbb{N}\rightarrow\Bbb{C}$ and $T_f\psi=f\psi$. I want to prove the following statement: Suppose ...
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78 views

What is Mountain Pass theorem?

The title says it all, what is mountain pass theorem, how does it work is variational problems with doubly differential constraint?
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36 views

A question about a relationship of expressions got from change of variables/inner products

Suppose $F:L^2(S) \to L^2(T)$ is linear homeomorphism such that $F(v) = v \circ \mathcal{F}$ where $\mathcal{F}:T \to S$ is a diffeomorphism. Suppose $$\lVert F(v) \rVert_{L^2(T)} \leq C\lVert v ...
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101 views

Finding the spectral radius and spectrum .

I am solving the following question : If $k:[0,1]^2\to \mathbb C$ is continuous and $T_k : C[0,1] \to C[0,1]$ such that $$(T_kx)(t)=\int_0^t k(t,s)x(s) ds$$ Define $k_n: [0,1]^2\to \mathbb C$ ...
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85 views

Is the trivial vector space considered a Banach space?

Is the normed vector space $\{0\}$ considered a Banach space? I am asking this question because Cartan's Differential Calculus implicitly assumes that the identity operator has norm 1 in Section 1.9. ...
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23 views

An example of a domain where no Trace can be defined .

Let us take for example a domain $\Omega \subset \mathbb R^2 $ and $\Omega=B_r(0)$ \ $\{(0,y) :-r <y<0\}$ , why should i not be expecting the trace of a function $u \in H_p^m(\Omega)$ . And ...
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112 views

Lesbegue decomposition of sum of Borel measures.

Rudin asked: Suppose $\mu_{n}$ is a sequence of positive Borel measures on $\mathbb{R}^{k}$and $$\mu(E)=\sum^{\infty}_{n=1}\mu_{n}(E)$$Assume $$\mu(\mathbb{R}^{n})<\infty$$Show that $\mu$ is a ...
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53 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
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189 views

convolution of L1 function with a harmonic oscillation

I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, ...
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139 views

$f$ is concave and convex in its arguments.

Suppose that X be a reflexive Banach space and function $f:X×X↦R$ which is concave in its first argument and convex in its second one. How to prove $f(x,x)=0$ for all $x∈X$?
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78 views

Properties of the spectrum

Let $\rho$ denote the resolvent of a closed operator and if $\lambda \in \rho(A)$, define $R(\lambda,A) := (\lambda I -A)^{-1}$. If $\mu$ is sufficiently small ...
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30 views

Density of finite element functions in $W^{1,p}(\Omega)$

I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of ...
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144 views

Unbounded operator on $C[0,1]$

It is well known that the differential operator is an unbounded operator on the space of all continuously differentiable function on $[0,1]$. However,I found difficulties in finding an unbounded ...
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83 views

Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out. Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ ...
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108 views

Sampling Theorem Poisson Formula

Theorem If the Fourier transform $\hat{f}(w)$ of a signal function $f(x)$ is zero for all frequencies ouside the interval $-w_c\leq w \leq w_c$, then $f(x)$ can be uniquely determined from its sampled ...
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55 views

If $v_n \to 0$ in $L^2$, does $v_n \to 0$ in $L^8$?

If $v_n \to 0$ in $L^2$, does $v_n \to 0$ in $L^8$? Suppose the domain is a compact surface in $\mathbb{R}^n$. For example it could be a sphere.
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86 views

Laplace equation

I've been asked to compute the fundamental solution (in the distributional case) of the Laplacian. I have reach $u(r)=c\cdot r+c'$ for $r>0$ but i don't know how to prove that $\Delta ...
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113 views

Kernel Function

Can somebody give me an institution of Kernel function and why we define it in Hilbert Space and what does it means that Kernel function maps from one Hilbert space to another Hilbert space ?
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128 views

An inequality involving exponential of compact self-adjoint operator (SOLVED)

I've run into a tricky functional analysis problem. Here it is: Suppose $$A: H \to H$$ is a compact self-adjoint operator on a Hilbert space H. Assume that the spectrum of $A$ is located in the open ...
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218 views

A continuous embedding.

If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
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385 views

Learning Functional analysis in one month.

Background: Master in Engineering, applied physics. Now: New phd student in applied math. Goal: Pass functional analysis with good grades. I have not had so much time for studies period since there ...
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63 views

A reference to basic properties of compact operators without assuming completeness

I am looking for a textbook showing that (i) every compact operator is bounded and (ii) composing a compact operator with a bounded one (and a bounded operator with a compact one) gives a compact ...
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95 views

About Lebesgue measure

This is a problem of Lebesgue measure and measure theory specifically. Suppose that $f:\mathbb{R}^2\longrightarrow [0,\infty)$ is measurable. $\Omega_1\subseteq \mathbb{R}^2$ is Lebesgue ...
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70 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
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356 views

Dirichlet and Neumann boundary conditions Finite Element Method

I have the following problem in Finite Element Method $$ -(\alpha u')' + \beta u' + \gamma u$$ with $ \Omega = (0, 1)$, $ u(0) = 0 $ and $ u'(1) = 3 $ to be able to write the weak formulation of ...
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149 views

A weak convergence in Sobolev space.

Let $u \in C^0([0,T], H^{s-1}(\Bbb R^n)). $ Let $\{t_n \} \subset [0,T]$ such that $\lim_{n \to \infty} t_n = t_0$. Let $ u(t_n ) \to u(t_0) $ in the Sobolev space $H^{s-1} ( \Bbb R^n )$ for $s = ...
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421 views

Give an example a function sequence in the Schwartz space $\mathcal S(\Bbb R)$ which does not converge

Give an example of a function sequence in the Schwartz space $\mathcal S(\Bbb R)$ which does not converge. That is, for any $a,b \in \Bbb Z_+$, $$ \|f_n\|_{a,b} < \infty, $$ but $$ ...
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85 views

Continuity of a Linear map $C^\infty_c(K) \to X$

Let K be a compact set. How does one show the following? If a linear map $T:C^\infty_c(K) \to X$ into a normed vector space X is continuous then there exists $k \geq 0$ and $C>0$ such that ...
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66 views

Positive maps on $\mathcal{B}(\mathcal{H})$ to itself

Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
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266 views

Describe dual space of $C[0,1]$

I am stuck in this problem. Describe the dual space of $C[0,1]$, where $C[0,1]$ is the Banach space of all real continuous functions on $[0,1]$ induced norm $$ \|x\|_{\max}=\sup_{t\in ...
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381 views

Reflexive Banach Spaces

How do I show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive without using any deep functional analysis theorems?
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26 views

Series function help

I want to find a function such that $$ \sum_{0<j<n/k } f(kj)=1 $$ Where the sum j is taken over the natural numbers, And the series is satisfied for all integers k and n, I was thinking of ...
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235 views

Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$

Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which $$ \lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v. $$ But ...
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63 views

Differentiation of the norm

Let $f = f(t,x) = f(t, x_1, \cdots , x_n) . $ If $f \in C^1 ([0,M],W^{s,2}(\Bbb R^n))$ (which means that $g(t) :=\| f \|_{W^{s,2}(\Bbb R^n)}$ is differentiable on $[0,M]$ and $\frac{d}{dt} g(t)$ is ...
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73 views

The Equivalents Norms in Sobolev Spaces

Show that over $W^{m,p}(a,b)$ the norms $$||w||^p_{W^{m,p}(a,b)} = \sum_{i=1}^m\int_a^b|w^{(i)}|^pdx$$ $$||w||^p_{W} = \int_a^b|w|^pdx + \int_a^b|w^{(m)}|^pdx$$ are equivalents. pdta: By ...
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48 views

geometric charaterization of complex interpolation spaces $(H,Y)_\theta$ where $H$ is a Hilbert space?

Let $C$ be the class of Banach spaces $X$ such that there exists $0<\theta<1$, a Hilbert space $H$ and a Banach space $Y$ such that $$ X=(H,Y)_\theta $$ (complex interpolation of Calderon). ...
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136 views

Separation in infinite dimensional normed space

I would like to construct some counterexamples: $E$ is an infinite dimensional normed space. Let $C, D$ be nonempty convex subsets in $E$ such that $$ C\cap D=\emptyset. $$ There is no vector $f\in ...
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136 views

i am stuck with functional analysis

Let Ɗ be a convex w*-compact subset of Ӿ*, where Ӿ is a separable normed space. Show that the external boundary of ∂Ɗ of Ɗ can be written as a countable intersection of open subset of Ɗ known as Gδ
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45 views

An estimation using the functions in Schwarz class.

Define $$ U := C^0 ([0,T], W^{1,2} ) \cap C^1 ([0,T] \cap L^2 ) \cap L^\infty ([0,T] , W^{s,2} ).$$ Then how can I prove that $ \lim_{x_j \to \infty}| u |^2 = 0 $ by using the fact: $$ C^1 ([0,T] , ...
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95 views

Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...