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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124 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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122 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in H^...
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71 views

Semigroup of operators: weak continuity at 0+ implies weak continuity at any t > 0

Let ($E$, $d$) be a metric space. Consider the semigroup $\{P(t)\}_{t\geq 0}$ of bounded linear operators on the Banach space $\hat{C}(E)$ of continuous real functions on ($E$, $d$) vanishing at ...
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1answer
232 views

weak* convergence for sequence in $ L^\infty$

Let $ \Omega \subset \mathbb{R}^d $ be a bounded and open set. Suppose $ \{f_n\} \subset L^{\infty} (\Omega) , f \in L^{\infty} (\Omega) $. Prove that $ f_n \rightharpoonup^* f \ \ \text{in} \ \ L_{...
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120 views

Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
5
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3answers
409 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any $|\...
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1answer
50 views

how to find norm of the operator Ax(t) = cos (t)*x(t)?

Please explain me how can I get norm of this operator: $Ax(t)=\cos(t)x(t)$ where $A\colon C[-\pi/2,\pi/2] \to C[-\pi/2,\pi/2]$. Thanks
5
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3answers
86 views

Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions

An interval in this problem may be open, closed or half open. We regard a singleton $\{a\}$ as being an interval also. A step function is a real valued function on $\mathbb{R}$ which is a linear ...
0
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1answer
63 views

Trace map from $H^1$ into $H^{\frac 12}$, does this statement imply another?

Consider trace map $T:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ on a sufficiently smooth domain $\Omega$. It has a partial inverse $E$. If we have the statement $$F(u,Eu) = 0\quad\text{for all $...
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1answer
153 views

Determinant: Continuity

Reference Build-up on: Determinant: Definition Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its determinant $\det:\mathcal{L}(V)\to\mathbb{C}$. Introduce a norm $...
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2answers
117 views

C*-algebras: Literature?

I'd like to better understand states on C*-algebras. What properties should I investigate and in which order? (Positive functionals, extremal states, Schwarz's inequality, Kadison's inequality, what ...
3
votes
1answer
58 views

Help with proof that $H^{1}(\mathbb{R})$ is closed under multiplication.

Edit: Prove that if $u,v \in H^{1}(\mathbb{R})$ then $uv \in H^{1}(\mathbb{R})$. My idea is to approximate with functions in $C^{\infty}(\mathbb{R})$ with compact support. Let $u,v \in H^{1}(\...
3
votes
1answer
98 views

$c_0$ is not isometric to $c_0 \oplus c_0$

$c_0$ is the Banach space of sequences converging to zero and $c_0 \oplus c_0$ is its algebraical direct sum with itself equipped with the norm $\|(\xi,\eta)\| := \|\xi\|+\|\eta\|$. How to prove that ...
2
votes
1answer
33 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since $\Phi$...
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5answers
276 views

Determinant: Alternative Definitions

Reference Foundation for: Determinant: Continuity Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. The rank of an endomorphism: $$\mathrm{rank}T:=\dim\left(\mathrm{im}T\...
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38 views

Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
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2answers
219 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Problem 8, Section 2.7

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
5
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1answer
513 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.7, Problem 9

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or complex-...
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1answer
51 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
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2answers
46 views

Knowing a function while only knowing its partial derivatives?

So again we study a physics course without studying mathematics course We are in the work energy chapter , and I'd like to know if you can know the function $f(x,y,z)$ if you know all of its partial ...
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1answer
53 views

$C_c(\mathbb R^n)$ is not dense in $\mathcal L^\infty(\mathbb R ^n)$

I'm having some difficulties in manipulating the space $\mathcal L^\infty(\mathbb R ^n)$, and I want to show that $C_c(\mathbb R^n)$ is not dense in $\mathcal L^\infty(\mathbb R ^n)$, but I can't find ...
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1answer
58 views

Exercise on functional analysis

I have a problem with this exercise of functional analysis about weak convergence: Let $U:=\prod_{i}^{d}(a_{i},b_{i})$ a subset of $\mathbb{R}^d$ with $a_{i}<b_{i}$ for each i and let $f\in{L^{p}(...
3
votes
0answers
140 views

Separable Hahn-Banach and the axiom of choice

We had in our functional analysis course a proof for the Hahn-Banach theorem on a separable Banach space which doesn't need, according to our professsor, the axiom of choice. Yesterday I read the ...
2
votes
1answer
243 views

Functional Derivative (Gateaux variation) of functional with convolution

I have the following functional \begin{align*} F[f]=\int f(x) \log(g(x)) dx$ \end{align*} where $g(x)$ is given by convolution $g(x)=y(x) * f(x)=\int y(\tau) f(x-\tau) d\tau$, so \begin{align*} F[...
2
votes
1answer
101 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), i=...
0
votes
0answers
36 views

The resolvent of a differentation operator on $C[a,b]$

Consider a densely defined operator $A : C[a,b ]\rightarrow C[a,b ]$, $$Au=u^{\prime}$$ with domain $$D(A)= \{ u\in C^1[a,b]: u(b)=ku(a) \}$$ for some $k>0$. I have to find $R_A(\lambda)$ for $...
0
votes
1answer
126 views

Fundamental theorem of calculus with Gâteaux differentials and Riemann integrals

Let $f:[a,b]\to E$ where $E$ is a Banach space and let $Df(x,h)$ be its Gâteaux differential in $x$ with direction $h$. If $\mathbb{R}\to E$, $h\mapsto Df(x,h)$ is linear and continuous, then we write ...
3
votes
1answer
83 views

Having trouble understanding a proof after it applies the Hanh Banach theorem.

I have been reading a proof on the convergence of Newton's method that has been fairly easy to follow except for a single step that has totally mystified me because it suddenly depends on a lot more ...
1
vote
1answer
40 views

Homeomorphism from $K$ to $\Phi_{C(K)}$

Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that $\...
3
votes
2answers
123 views

Metrizability of the unit ball $B_{X^*}$.

I am trying to prove the assertion: If $X$ is a separable normed space, then the unit ball in $X^*$ with the weak* topology, $(B_{X^*},\sigma(X^*,X))$, is metrizable. Firstly, I took $D=(x_n)\...
2
votes
1answer
111 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
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votes
0answers
46 views

Spectrum of a bounded operator on a (not necessarily Banach) normed vector space

It's well known that on a Banach space, the spectrum of each bounded operator is compact in $\mathbb C$. What about a general normed vector space? Is there a counterexample if we don't assume ...
3
votes
1answer
149 views

Closed subspace. A Hahn–Banach theorem consequence

I am trying to prove: If M is a subspace of a normed space $X$, that $\overline{M}=\bigcap\{\ker(\phi):\phi|_{M} = 0 \}$ It is really easy to see that $\overline{M} \subset \bigcap\{\ker(\phi):\phi|...
2
votes
0answers
67 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I noticed,...
2
votes
0answers
66 views

Resolvent operator

Let's consider the following operator on $L^2(\mathbb{R}^3)$ $$A(t)=\Delta+b(t,x)\cdot\nabla$$ where $\Delta$ is the Laplace operator and $b(\cdot,\cdot)$ a smooth vector field. How to compute the ...
2
votes
1answer
71 views

If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

The following is a remark of Murphy's C*-algebras and operator theory: . I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is ...
4
votes
0answers
70 views

The weak-star topology is Completely Hausdorff (in particular Hausdorff).

Let $X$ be a normed space, $X^*$ its dual space, $(X^*, w^*)$ is completely Hausdorff. Proof: Let $f, g \in X^*$, $f\neq g$ then $\exists x\in X$ such that $f(x)\neq g(x)$ i.e. $\hat{x}(f)\neq \hat{...
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1answer
535 views

Smallest constant of Lipschitz retraction from bounded to continuous functions

Let $B$ be the space of all bounded functions $f:[0,1]\to\mathbb R$ equipped with the supremum norm*. It contains $C$, the space of continuous functions on $[0,1]$, as a subspace. An $L$-Lipschitz ...
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1answer
67 views

calculate weak derivate of $|x-2|^2$

Let $u$ be a function with $u(x):=|x-2|^2$ on $I:=(-1,1)$. I want to test whether $u \in H^2(I) \backslash H^3(I)$. Let $\phi$ be in $C_0^\infty(I)$. Then: $T_u(\phi '') = \int_{-1}^1 |x-2|^2 \...
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1answer
52 views

the norm of a linear operator

In in the demonstration of Lax-Milgramm lemma, they use a linear operator $A:V\to V$, where $V$ is a Hilbert space; My basic problem is how to prove that $$\|Au\|_V=\sup_{v\in V^*}\frac{(Au,v)_V}{\|...
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0answers
52 views

Confusion about domains in PDE and “every $C^k$ manifold can be thought of as a smooth manifold”

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like ...
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vote
0answers
56 views

Newton method for maps between Banach spaces

I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here): Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a ...
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votes
1answer
287 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.7, Problem 6

Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$. ...
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votes
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233 views

defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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vote
0answers
61 views

Can a Local Fractional Differential Operator exist?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$. The derivative of $f$ is defined pointwise, and we say that $f$ is differentiable if the derivative exists in each point. Higher order derivatives are ...
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1answer
104 views

If the dual unit ball of a normed space $X$ is metrizable in the weak-$*$ topology then $X$ is separable

Let $X$ be a normed space and $(B_{X^*},w^*)$ be the unit ball of the dual space $X^*$ endowed with the weak-$*$ topology. Here is a proof a the fact that if $(B_{X^*},w^*)$ is metrizable then $X$ is ...
0
votes
1answer
48 views

Dynamics: Schwinger-Dyson-Expansion

Given a C*-algebra $\mathcal{A}$ Consider a free generator $\delta_0:\mathcal{D}_0\to\mathcal{A}$ with $\overline{\mathcal{D}_0}=\mathcal{A}$. Introduce a perturbation $\delta_V:\mathcal{A}\to\...
1
vote
1answer
133 views

Schwartz space on $\mathbb T^{n}$

For the definition of Schwartz space space on $\mathbb R^{n},$ see this. My Questions: (1)Is it make sense to talk of Schwartz space on torus $\mathbb T^{n}$ ? If yes, what can be the analogous ...
0
votes
1answer
79 views

Lower bound for the norm of the resolvent

I need to prove next statement (I want to do it for general case) $\|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}$ I think it could be like this let $a\in \sigma(A) z \...
5
votes
0answers
85 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...