Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
10,143 questions
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Converse of the spectrum theorem for compact operator [closed]
I am looking for an example of non-compact linear operator A on Hilbert space such that the spectrum of A is a sequence of eigenvalues of finite multiplicity converging to 0.
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Asymptotic behavior of the Green function of $(-\Delta_{g_{\mathbb{S}^3}}+1)^{1/2}$ on $\mathbb{S}^3$
Let $A=\left(-\Delta_{g_{\mathbb{S}^3}}+1\right)^{1 / 2}$. I expect the Green function on $(\mathbb{S}^3,g_{\mathbb{S}^3})$ has the asymptotic behavior $G(x, y) \sim c d(x, y)^{-2}$ as $x \rightarrow ...
- 569
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Countable non-relatively compact subset of $C_p(X)$ of nonnegative functions with a limit point
It is known that for a compact Hausdorff space $X$, the space $C_p(X)$ of continuous real-valued functions on $X$ with the topology of pointwise convergence is angelic. In particular, $C_p(X)$ has ...
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Lebesgue differentiation theorem and convolution with measures
A version of the Lebesgue Differentiation theorem states that if $f \in L^1_{\mathrm{loc}}(\mathbb{R}^n)$, then
\begin{equation}
\lim_{r \to 0} \frac{1}{|B_r(x)|}\int_{B_{r}(x)} f(y) \, \mathrm{d}y = ...
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Forgetting about topology on quantum groups?
For the category of locally compact groups we know that there is a forgetful functor
$$\mathbf{LocCptGrp}\longrightarrow\mathbf{DiscreteGrp} $$
which sends $G$ to $G_d$, the discrete version of $G$. ...
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74
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Linear independence of shifted functions
Let $a \neq 0$ and $0 < b < 1$ be two real numbers. Consider two operators on the space of functions $\mathbb{R} \to \mathbb{C}$ or $\mathbb{R} \to \mathbb{R}$ given by $Af(x) = f(x-a), Bf(x) = ...
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1-unconditional basis for the quotient subspace (II)
By Propositions 1.a.9 and 1.a.11 in [1], the following assertion holds true: Let $E$ be a Banach space with a normalized unconditional basis $(e_n)^\infty_{n=1}$. Then there exists a $\lambda>1$ ...
6
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129
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Non-equivalent norms in the Picard–Lindelöf fixed-point argument
In the standard proof of the Picard–Lindelöf theorem for the existence and uniqueness of solutions to the initial-value problem
$$y'(t) = f(t,y(t)), \quad y(t_0)=y_0,$$
one typically works in the ...
- 165
1
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1
answer
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1-unconditional basis for the quotient subspace
Question. Let $E$ be a super-reflexive Banach space with a normalized 1-unconditional basis (or alternatively, let $E$ be a subspace of $L_p[0,1]$ with $1<p<\infty$
that satisfies the above ...
9
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1
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390
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On the proof of an invariant Hahn-Banach theorem
In Rudin's book Functional Analysis there is something I do not understand in the proof of the following theorem
(theorem 5.24 in this book):
Suppose $Y$ is a subspace of a normed linear space $X$, $f ...
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75
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Does the additive channel model arise from average sum power constraint?
The Additive White Gaussian Model ($\mathsf{AWGN}$) model is the following: You send a message $x$ from a finite set of real alphabets $\chi$ and White Gaussian Noise (noise of Gaussian distribution $\...
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2
votes
1
answer
54
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Power type $p$ modulus of asymptotic uniform smoothness under $\ell_{p}$-direct sum
Now let us recall the definitions of asymptotic properties of Banach spaces.
Let $X$ be a Banach space and $t>0$, the modulus of asymptotic uniform smoothness of $X$ is defined by
$$\overline{\rho}...
- 101
3
votes
1
answer
168
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Stability of the Radon-Nikodým property under infinite sum
I would like to ask a question concerning the stability of direct sums of Banach spaces with the Radon-Nikodým Property (RNP). As far as I have reviewed the literature, relevant results are presented ...
- 101
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Criteria for the completion of an ultrabornological locally convex vector space to remain (ultra)bornological
In what follows, all vector spaces are over the field of scalars $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. One has as basic facts in functional analysis that:
Any bounded linear operator $T:E\...
- 9,158
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Does there exist some result about vector-valued non-commutative measure spaces?
It is well-known that for any Banach space $X$, and a probability space $(\Omega,\Sigma,P)$, we may define a $X$-valued measurable function space $(\Omega,\Sigma,X)$. And a von-Neumann algebra $\...
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