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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Set of diffeomorphisms on a manifold

It is well known that given a compact smooth boundaryless manifold $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r \geq 1$, is open in $C^{r}(M)$, the set of continuous functions (for ...
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76 views

Reflexive Banach spaces and norms

Let $(X,|.|)$ be a reflexive Banach space, and $Y\subset X$ such that $(Y,|.|_Y)$ is a Banach space with a norm $|.|_Y$ stronger than $|.|$, i.e. there's a constant $C$ such that $$|y|\leq C |y|_Y, \ \...
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51 views

Characterization of nowhere differentiable functions

Let $N:=\{f\in C([0,1])\vert \text{ f is nowhere differentiable } \}$ and $A_n = \{f\in C([0,1]) \vert \exists x\in [0,1]s.t. \forall y\in[0,1]: |f(x)-f(y)|\leq n |x-y|\}$. Now I have already ...
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15 views

Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense for $M$ a compact Riemannian manifold?

Let $M$ be a compact Riemannian manifold. Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense?
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159 views

Reflexive normed spaces are Banach

I want to prove that a reflexive normed space $X$ is a Banach space. By the definition of the reflexive space, the evaluation map $J:X\to X''$ is a bijection. All I need is to prove that the ...
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44 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 $2$-norm-...
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108 views

An application of Hahn-Banach (separation) theorem

Here is a quotation of a book: Let $S(A)$ denote the state space of a C*-algebra $A$ and $M\subset S(A)$ denote a weak-$*$ closed convex set. Assume there is a state $\psi$ which does not belong ...
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62 views

Pervin quasi-uniformity

for a subset $ A$ of a set $X$ we set $S_{A} =[(X - A)×X]∪ [X ×A]$. We recall that if $X$ is a topological space, then the Pervin quasi-uniformity of $X$ is generated by the subbase $\{S_{G}: G \...
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28 views

Verification of a contraction

Let $A\colon \text{dom}(A) \to \mathcal{H}$ be a densely defined symmetric operator on a Hilbert space $\mathcal H$. The symmetry implies that $$ \|(A + i)f\|^2 = \|Af\|^2 + \|f\|^2 \quad \text{for ...
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76 views

Find norm of operator

I have a linear functional $$A: L_2[0,2] \to \mathbb R, Ax = \int_0^2(t^2+2)x(t)dt$$ I need to find $C$, trying to measure $C$ and $||Ax||$ to find it, but how can I do it in this problem?
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1answer
56 views

Extending a functional with same norm

I have: $X = <\mathbb R^2, ||(x,y)|| = \sqrt{4x^2+y^2}>, L = \{(2x,3x), x \in\mathbb R\}, $ $\phi_0 \in L^* : \phi_0 (2x,3x) = -2x$ I need to extend $\phi_0$ on $X$ without changing norm (...
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1answer
20 views

Density of bounded functions in $L^1(0,T;L^1(M))$?

Let $u \in L^1(0,T;L^1(M))$ where $M$ is a compact Riemannian manifold. Is it possible to find $u_n$ such that $u_n \to u$ in $L^1(0,T;L^1(M))$ and $u_n$ are bounded everywhere or almost everywhere on ...
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89 views

Find a norm of operator form l1 to l1

I have an operator $A: \ell_1 \to \ell_1, Ax = (x_1+x_2, x_1-x_2, x_3,...,x_k,...)$ AFAIK, norm of $\ell_1$ is $\sum_{n=1}^{\infty}|x_n|$ How to find a norm of this operator?
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110 views

Gaps in my proof of the Arzela-Ascoli Theorem - help and expertise greatly appreciated for an alternate formulation.

I have a general outline of the proof of the Arzela-Ascoli Theorem but have trouble filling in the gaps of the theorem. I have posted the entire general method I believe to be correct below. I was ...
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1answer
88 views

Find the norm of the operator $A:L_2[0,2] \rightarrow L_2[0,2]$ defined by $(Ax)(t) = t \, \operatorname{sgn}(t-1) x(t)$

I have operator: $\boldsymbol{L}_2[0,2] \to \boldsymbol{L}_2[0,2], ( Ax)( t ) = \boldsymbol{t} \operatorname{sgn}(t-1)x(t)$ I need to find operator norm or say that operator isn't bounded. ...
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211 views

A topological vector space with countable local base is metrizable

I feel confused by the proof of the following theorem in Rudin 2/e: Theorem 1.24 If $X$ is a topological vector space (t.v.s.) with a countable local base, then there is a metric $d$ on $X$ s.t. ...
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1answer
58 views

Nontrivial functionals on $l^\infty$ vanishing on $c_0$

I understood that the dual of $c_0$ is a proper subspace of the dual of $l^\infty$, by Hahn-Banach theorem. But how can I find functionals in $(l^\infty)^*$ vanishing on $c_0$?
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116 views

Maximal norm of tensor product

Definition 3.3.3. (Maximal Norm) Given $A$ and B (two C*-algebra), we define the maximal C*-norm on $A \odot B$ to be $$||x||_{max}=sup\{||\pi(x)||: \pi: A\odot B\rightarrow B(H) ~a~*-homomorphism\}$$...
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1answer
110 views

Weak convergence on Banach space

Let $(X,|.|)$ be a Banach space, and $Y$ is a linear subspace of $X$ which is dense in $X$. Now if we have another norm $|.|_Y$ in $Y$ which is comparable to $|.|$ by $$|y|\leq C |y|_Y, \ \ \ \forall ...
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1answer
49 views

The closure of the range of the operator of a symmetric nonnegative operator on a real separable Hilbert space $H$?

If $Q$ is a symmetric nonnegative operator on a real separable Hilbert space $H$. We have known that an orthonormal set $\{ e_i \}$ can be chosen such that $Q e_i=\lambda_i e_i, \lambda_i > 0$...
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213 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = \frac{1}{\sqrt{2\...
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203 views

Measure theory questions applied to Second Order PDE

Most of the questions are more measure theory and integration related but I need to give some context, so I will now. Consider the quasilinear 2nd-order partial differential equation $$-\text{div}(a(...
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81 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and $L_p(\...
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1answer
43 views

Inequality $||f-g|| < \epsilon \Rightarrow |E[f] - E[g]| < \epsilon$

Let $C(X)$ be the space of continuous bounded functions on some metric space $(X,d)$. Can it be shown that if $||f-g||_\infty < \epsilon$ if follows that $| \int f \, \text{d}P - \int g \, \text{d}...
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145 views

Uncountable disjoint union of measure spaces

Let $(a,b)$ be an interval. Let $(A_i, \Sigma_i, \mu_i)$ be a measure space for each $i \in (a,b)$. Is it possible to put a measure space on the disjoint union $$\bigcup_{i \in (a,b)}\{i\}\times A_i?$...
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23 views

What is the definition of $\min$, $\max$ of functions, $f_i$?

I have a couple of questions: What is the definition of the expressions on the right-hand side? Each $f_i : X \to \overline{\mathbb{R}_+}$ $$ h = \max(f_1, f_2, f_3,...f_n) $$ $$ h = \min(f_1, f_2, ...
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83 views

Question about mollifiers.

So here is my problem, Let $\rho \in C^\infty (\mathbb{R}^n,R)$ with $\rho\geq 0$, $\rho(x)= 0 \; \forall \|x\|\geq 1$ and $\int_{\mathbb{R}^n}\rho(x)dx=1$. Further, consider the linear map $K_f:L^p\...
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64 views

how does semi-inner product and symmetric positive semi-definite bilinear form are different?

Given vector space $V$ over scalar field $\mathbb R$, I wonder if two definitions "semi-inner product" and "symmetric positive semi-definite bilinear form" are actually equivalent. The definition of "...
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182 views

Euler's Refutation of Fermat's Conjecture

Fermat postulated that all numbers of the form $$2^{2^n}+1$$ are prime (where n = any integer). Then Euler came along with a rather ingenious proof that this was not, in fact the case. I came across ...
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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 =\int_{-1}^...
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1answer
37 views

How to verify $H\otimes K \cong \bigoplus\limits_{i\in I}H$

Let $H,~K$ be the Hilbert space. if $\{v_{j}\}_{j\in J}\subset H$ and $\{w_{i}\}_{i\in I}\subset K$ are the orthonormal bases, then how to construct the isomorphic mapping: $H\otimes K \rightarrow \...
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45 views

Question about Hilbert-Schmidt Operators

I have already tried but I failed. I can't show it is. I used this way: $| (Kf)_n |^2 \leq c_n \|f\|^2$, and therefore $\|Kf\|^2 \leq \|c\| \|f\|^2$, so that $\|K\| \leq \sqrt{\|c\|} $ is bounded, ...
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1answer
37 views

$*$-homomorphism of the tensor product

Let $A,~B,~C,~D$ be the C*-algebras and the "$\odot$" denotes the algebraic tensor product. Proposition 3.1.16 (Tensor product morphisms). Given $*$-homomorphisms $\phi: A\rightarrow C$ and $\psi: B\...
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66 views

Weak convergence and strong convergence in $L^1$.

Suppose that $\Omega$ is a Lebesgue measurable set,$f_n \rightharpoonup f$ in $L ^1(\Omega)$ and $\|f_n\|_{L^1(\Omega)}\rightharpoonup\|f\|_{L^1(\Omega)}$, then can I say that $f_n → f$ strongly in $L^...
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1answer
2k views

Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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1answer
80 views

Basis of an infinite dimensional Banach space

Can somebody please check the correctness of this proof, since I am new to this? Thank you in advance. Given $X$ a normed space and $Y$ a proper subset of $X$ that is a linear subspace, prove that $...
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Schrödinger-Operator on $L^2[0,2\pi]$.

In Reed-Simon Analysis of Operators they often talk about operators like $H = - \Delta +V$ as an operator on $L^2[0,2\pi]$ (like in Theorem XIII 88. What do they mean by that? Or is their a canonical ...
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If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
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232 views

continuous extension and smooth extension of a function

Let $X$ be a metric space. Let $E$ be a subset of $X$. (1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such ...
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47 views

Closed and bounded subsets of twice continuously differentiable functions

Can you show that a closed and bounded set $A$ of $C_{2}[0,1]$ is has a compact closure in $C_{1}[0,1]$? I think the Arzela theorem must be invoked here, and it suffices to check that $A$ is ...
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1answer
113 views

Weak convergence: equivalence of definitions

Consider a sequence of random variables $(X_n)_{n\geq 0}$ and a random variable $X$. How to prove that the two following definitions of weak convergence are equivalent? Def 1 $(X_n)_{n\geq 0} \...
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1answer
328 views

Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
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71 views

Necessity of hypothesis in distance from a set in an inner product space

In Kreyzig's Functional Analysis book, one theorem in inner product spaces is about the existence and uniqueness of a minimal point from a set. 3.3-1 Theorem (Minimizing vector). Let $X$ be an ...
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1answer
34 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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29 views

Is the linear operator $T_2^{-1}T_1 :U_1 \to U_2$ bounded if $T_1\in L(U_1,H)\ \ T_2 \in L(U_2,H)$ $\mathrm{ker}\ T_2=\{0\} $?

Let $U_1, U_2, H$ are Hilbert spaces. $T_1\in L(U_1,H)\ \ T_2 \in L(U_2,H)$, $\mathrm{ker}\ T_2=\{0\} $, and the image of $\ T_1,T_2$ are the same, i.e. $\mathrm{Im}\ T_1=\mathrm{Im}\ T_2$, My ...
2
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1answer
133 views

$L^\infty$ bound on solution of $u_t -\Delta u =f$ where $f \in L^\infty$ and $u_0 \in L^\infty$??

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some boundary conditions. Consider the weak form $$\int_{\Omega}u_t\...
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3answers
1k views

The difference between hermitian, symmetric and self adjoint operators.

I am struggling with the concept of hermitian operators, symmetric operators and self adjoint operators. All of the relevant material seems quite self contradictory, and the only notes I have to go ...
8
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2answers
600 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: $$\{...
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1answer
47 views

Linear onto isometry.

Let $K$ and $L$ be two compact set and $T$ is an linear onto isometric from $C(K)$ to $C(L)$. My question is that $T(1)$ is the identity map in $C(L)$, where 1 is the identity map in $C(K)$ . give me ...
4
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1answer
93 views

The Levi-Civita connection in infinite dimensions

Is there an analogue of the Fundamental Theorem of Riemannian Geometry for (some subclass of) infinite-dimensional manifolds?