All Questions
Tagged with fa.functional-analysis hilbert-spaces
467 questions
5
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1
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Inner product of linear bounded operators between Hilbert spaces
Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\...
- 5,327
5
votes
1
answer
206
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Compactness in trace class operators space
Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$.
Are there easy ...
- 59
5
votes
1
answer
1k
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Left and right eigenvectors are not orthogonal
Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and ...
- 73
5
votes
2
answers
148
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Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$.
Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
5
votes
1
answer
229
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Canonical multiplication representation of self-adjoint operator in quantum chemistry and coding theory research
In my applied math research group, we are studying and going over functional analysis results from papers and theses from our institution to generalize their results and apply them in our discrete ...
- 620
5
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1
answer
197
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The largest topological copy of a Hilbert space contained in $\ell^1$
Let us consider $\ell^1$, the space of absolutely summable sequences in the space of complex numbers. Clearly every finite dimensional Hilbert space is topologically embedded into $\ell^1$.
...
- 4,058
5
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1
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872
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Besicovitch Almost Periodic Functions a subspace of what?
The common example of a nonseparable Hilbert space comes from the collection of Besicovitch almost periodic function spaces. Starting with $L^p_{\text{loc}}(\mathbb{R})$ we look at those elements ...
- 1,104
5
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2
answers
310
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Error estimate in the spectral theorem of compact operators on a Hilbert space
Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
- 2,246
5
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1
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179
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An extension of Lomonosov Theorem
Let $H$ be a complex infinite dimensional separable Hilbert space. There are various extensions of the following well known result:
Theorem (Lomonosov): Every nonscalar $T \in B(H)$ which commutes ...
- 1,175
5
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1
answer
578
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Infimum over all vector-valued L^2 spaces
Suppose I have a Banach space $E$ (which may be finite dimensional if you wish), a Hilbert space $H$ and a tensor $\tau \in H\otimes E$ in the algebraic tensor product. There are lots of ways to ...
- 18.7k
5
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0
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899
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Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?
Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases}
a_k,b_k\in \mathbb{R}\ \forall k=1,\...
- 181
5
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0
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116
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Multiplier algebra of Fock space
For any vector space, one may form the tensor algebra with multiplication being the tensor product. For a Hilbert space $\mathcal{H}$, the analogous construction is the Fock space
$$
\mathcal{F}(\...
- 439
5
votes
1
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381
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Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$
I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$.
Under which ...
- 734
4
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2
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433
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A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space
Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ ...
- 433
4
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1
answer
466
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Injection between non-isomorphic irreducible Hilbert space reps?
I must be missing something trivial here.
Let $G$ be, say, a reductive Lie group (or more generally any locally compact Hausdorff unimodular topological group). A unitary Hilbert space representation ...
- 41.4k
4
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3
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1k
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Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 1,284
4
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1
answer
273
views
Name for certain property of equivalent norms on finite-dimensional subspaces of a Banach space
Let $X=(X,\|\cdot\|)$ be a Banach space and suppose that $F\subset X$ is a finite-dimensional subspace. There is then an equivalent norm $|\cdot|$ on $F$ such that $|\cdot|$ is induced by an inner ...
- 201
4
votes
3
answers
728
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Inequality of von Neumann for more than two contractions
Good morning,
I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized ...
- 758
4
votes
1
answer
357
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Extending maps from dense $*$-algebras of $C^*$-algebras
Given $\cal{A},\cal{B}$ two dense $*$-algebras of two $C^*$-algebras $A$ and $B$ respectively, together with a $*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a ...
- 993
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2
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353
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Why $\lim_{n\to+\infty}\bigg(\bigg\|\sum_{f\in F(n,d)} A_{f}^* A_{f}\bigg\|^{\frac{1}{2n}} \bigg)\;\text{exists}?$
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.
For $A= (A_1,\cdots,A_d)\in\mathcal{L}(E)^d$ (not necessary to be commuting). Why
$$...
- 1,154
4
votes
2
answers
244
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Compact images of nowhere dense closed convex sets in a Hilbert space
Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...
- 41.8k
4
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2
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730
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Finite dimensional approximations of operators on Hilbert spaces
Let $e_1,e_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$. Let $A:V \to V$ be an operator. Finally, let $V_n = {\rm span}( e_1, \dots, e_n)$. Let $i_n : V_n \...
- 614
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1
answer
160
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If $F$ is a countably normed, nuclear Fréchet space, can I then find a fundamental system which exhibits both of these properties at once?
Let $F$ a Fréchet space.
This means that $F$ is a complete Hausdorff topological space whose topology can be generated by an increasing family of seminorms $\{ p_{n} \}_{n \in \mathbb{N}}$.
Let's ...
- 556
4
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1
answer
272
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On the automorphisms of the unitary group in the strong operator topology
Let $H$ be an infinite dimensional complex (or real) Hilbert space, and let $U(H)$ be the unitary (or orthogonal) group. We equip $U(H)$ with the strong topology.
Now, suppose that $\phi: U(H) \...
- 556
4
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1
answer
174
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A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 5,529
4
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1
answer
986
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RKHS norm of Lipschitz functions
Given a set $\mathcal{X}$ and RKHS $\mathcal{H}$ of functions on $\mathcal{X}$, we can recover a (pseudo)metric on $\mathcal{X}$ by $d(x,y)=||\phi_x-\phi_y||_{\mathcal{H}}$, where $\phi_x=k(x,\cdot)$.
...
- 101
4
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1
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384
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A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra
Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-...
- 1,396
4
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1
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386
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Invertible unbounded linear maps defined on a Hilbert space
It is well-known that, assuming the axiom of choice, there are unbounded linear maps defined not only on a dense subset but on all of Hilbert space. Is it possible that such a map is invertible?
- 3,873
4
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1
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1k
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RKHSs containing constant functions
Suppose $H$ is the reproducing kernel Hilbert space on a space $X$ with reproducing kernel $K$. If, say, $K - c$ is a positive definite kernel for some $c>0$ then $H$ contains the constant ...
- 11.4k
4
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1
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461
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On the self-adjoint part of a quasinilpotent operator
Disclaimer: this is not research-level, but I've read some non research-level questions/answers on quasinilpotent operators here, some of them involving renowned users. So I thought I'd give it a try. ...
- 660
4
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1
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301
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Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus
In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim ...
- 620
4
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2
answers
313
views
Do all unitary representations weakly converge to zero at infinity?
Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...
- 692
4
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1
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377
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Closure of the space of Fredholm operators
Let $X,Y$ be two Banach spaces.
A bounded operator $A$ is Fredholm if $\ker A$ and $\mathrm{coker} A$ are finite dimensional. Denote by $Fred(X,Y) \subset \mathcal{L}(X,Y)$ the space of Fredholm ...
- 2,533
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1
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Doubts on Reproducing Kernel Hilbert Spaces and orthogonal decomposition
I'm a CS student and I'm trying to learn RKHS theory to understand the passages made in this paper .
Among the bibliography I'm using there are "On the mathematical fundamentals of learning" and "...
- 151
4
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1
answer
110
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Graded adjointable operators on a graded Hilbert space
Given a graded Hilbert space $\mathbf{H} = \bigoplus_{k \in \mathbb{N}} \mathbf{H}_k$, one might extend the notion of adjoint to a "graded adjoint" defined as follows: $L \in B(\mathbf{H})$ is said to ...
- 969
4
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1
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151
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Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
- 12.5k
4
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1
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411
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Abstract Definition of a Reproducing Kernel Hilbert Space
This is a very basic question about the definition of a reproducing kernel Hilbert space (RKHS).
It seems the standard definition of a RKHS is as a Hilbert space $H$ of functions on some set $X$ ...
- 1,307
4
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1
answer
165
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Scattering of relativistic particle by long-range potential
Let
$\mathcal{H}=L^2(\mathbb{R}^3)$,
$H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)
and
$H=H_0+V(\vec{x})$
(where $V(\vec{x})$ is the operator of ...
- 552
4
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1
answer
196
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A kind of holomorphicity of maps on Hilbert space
Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?:
1)For every open set $U\subset H$ and every ...
- 356
4
votes
1
answer
127
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Proximal Operator image of convex functionals
Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator
$$
\begin{aligned}
&\Gamma_0\...
- 5,405
4
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1
answer
558
views
Weak topology on a pre-Hilbert Space
Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state.
Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product (...
- 5,529
4
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0
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152
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Maximally fine topologies on $B(H)$ making the unit ball compact
Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
- 190
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0
answers
111
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What is the native Hilbert space associated with the kernel $\frac{\sum \min{(x_i,y_i)}}{\sum \max{(x_i,y_i)}}$?
In this answer on MSE it is shown that the function
$$ K:(\mathbb{R}^{>0})^n\times (\mathbb{R}^{>0})^n\rightarrow\mathbb{R}\,\quad K(x,y)=\frac{\sum_{i=1}^n\min{(x_i,y_i)}}{\sum_{i=1}^n\max{(x_i,...
- 316
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0
answers
114
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Is this subspace of $B(\mathcal{H})$ known?
Let $\mathcal{H}$ be a Hilbert space. Suppose that I take a fixed ONB of $\mathcal{H}$ let us call it $\{ e_i \}_{i\in \mathbb{N}}$ and then I define
\begin{align*}
\|T \|_{D} = \sup_{l_i, m_i} \sum_{...
- 1,054
4
votes
0
answers
2k
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Eigenvalues and spectrum of the adjoint
In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$.
But in infinite dimensions this need no longer be ...
- 3,873
4
votes
0
answers
258
views
Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel
Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...
- 1,127
4
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0
answers
174
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Constant in trace theorem for balls
Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
user127450
4
votes
1
answer
128
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Closure of polynomials in $L^2_w$ with log-normal weight function
Consider the Hilbert space $L^2_w$ with scalar product $\langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution
$$ w(x)=\frac{1}{\...
- 43
4
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0
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164
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A modern reference for the "Intermediate Derivatives Theorem"
In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows:
Intermediate Derivative Theorem: Let $X\subset ...
- 1,247
4
votes
0
answers
185
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A strongly open set which is not measurable in the weak operator topology
Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
- 4,058