All Questions
Tagged with fa.functional-analysis fa.functional-analysis or
9,772 questions
12
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Direct proof of injectivity of $L_\infty$
I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.
$L_\infty$ as ...
- 1,697
12
votes
2
answers
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Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$
For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$.
Question
Given $\epsilon> 0$, find a "low-degree" ...
- 6,853
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votes
3
answers
881
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Bibliographic request concerning an article by Bernstein and Robinson
Concerning the article "Bernstein, Allen R.; Robinson, Abraham.
Solution of an invariant subspace problem of K. T. Smith and
P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in
finding ...
- 16.6k
12
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2
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1k
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A variation of the Ryll-Nardzewski fixed point theorem
Is there a fixed-point theorem that implies the following result?
Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...
- 45k
12
votes
3
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3k
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elementwise functions of positive definite matrix
The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) ...
- 96.4k
12
votes
4
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1k
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Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 11.1k
12
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2
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Where was/is Compensated Compactness used?
This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
- 261
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1
answer
1k
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What is the structure associated to almost-everywhere convergence?
Let $M(X)$ be the vector space (actually it's an algebra) of all equivalent classes of measurable functions $X\to \mathbb{C}$ (where $X$ is a measured space) modulo equality almost-everywhere.
One ...
- 549
12
votes
2
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948
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Banach space modulo a one-dimensional subspace =?
My question is the following:
Given an infinite dimensional Banach space $E$ and a one-dimensional linear subspace $F\subset E$. It is well-known that this one-dimensional linear subspace is closed ...
- 987
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5
answers
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Analogue of Cayley Hamilton theorem for operators on Hilbert space
Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
- 2,099
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1
answer
901
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Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem?
This post is an appendix of this one.
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is ...
12
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2
answers
605
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Who first defined locally convex topological vector spaces?
Who first defined the class of locally convex topological vector spaces?
- 2,655
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2
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847
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When is the closed unit ball in a smaller Banach space closed in a larger Banach space?
Recently I saw an interesting lemma:
For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in $L^...
- 5,169
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4
answers
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The image of a measurable set under a measurable function.
Let $f:X \rightarrow (Y, \mathcal{Y})$ be an abstract function, with $\mathcal{Y}$ a $\sigma$-algebra on $Y$. Endow $X$ with $f^{-1}(\mathcal{Y})$. Is then $f(X)$ a measurable set in $Y$? If not, are ...
- 5,407
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1
answer
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Sard's Theorem For Banach Spaces
Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...
- 2,099
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Locally constant functions with compact support = smooth ?
Hello,
I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.
Let F be a p-adic field. Then (from what I understand) $C_c^{\infty}(F)$ is the ...
- 231
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2
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647
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Do locally convex topological vector spaces embed into diffeological spaces?
The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results ...
- 35.5k
12
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1
answer
306
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Containment of $c_0$
I have the following question. I guess it's quite simple for experts.
Unfortunately, I could not come up with an answer yet.
Let $X$ be a Banach space which contains no copy of $c_0$.
Does it impply ...
- 225
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3
answers
870
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Measure theory in nuclear spaces
Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
- 8,512
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3
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2k
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Relevance of the complex structure of a function algebra for capturing the topology on a space.
This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is ...
- 3,699
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3
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Compactness of the set of densities of equivalent martingale measures
Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal P^\...
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Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
- 29.8k
12
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727
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A generalization of Rubio de Francia's inequality
Suppose that $\{I_m\}$ is a sequence of pairwise disjoint intervals in $\mathbb{Z}$. The well known Rubio de Francia's inequality says that for any function $f\in L^p(\mathbb{T})$, $2\le p<\infty$, ...
12
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1
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Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
- 6,853
12
votes
1
answer
229
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History of publication of von Neumann's characterization of orthogonally invariant matrix norms
Von Neumann has a result (rather well-known in convex analysis circles) which states that every orthogonally invariant matrix norm (meaning $\| P M Q\| = \| M \|,$ for any orthogonal $P, Q$) is a ...
- 96.4k
12
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1
answer
498
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Completely positive maps-equivalent definition
The most usual definition of the completely positive map (c.p.) between two C*-algebras (say, unital) is the following: $\sigma: A \to B$ should satisfy $\sigma(1)=1$ and for each $n \in \mathbb{N}$ ...
- 9,330
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1k
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Drawing conclusions by NOT using AC.
The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions defined ...
- 4,790
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What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 3,323
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1
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394
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Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 24.2k
12
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3
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891
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Notations for dual spaces and dual operators
I'm asking for opinions about the 'best' notations for:
1. the algebraic dual of a vector space $X$;
2. the continuous dual of a TVS;
3. the algebraic dual (transpose) of an operator $T$ between ...
12
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1
answer
2k
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Comparing Krein-Rutman theorem and Perron–Frobenius theorem
Krein–Rutman theorem is a generalization of Perron–Frobenius theorem, I know that things could be more subtle in infinite dimension, yet there's an important result in Perron–Frobenius that's missing ...
- 273
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3
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646
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Radii and centers in Banach spaces
Suppose I have a Banach space $V$ and a set $A \subseteq V$ such that for all $\epsilon > 0$ there exists $v$ such that $A \subseteq \overline{B}(v, r + \epsilon)$. Does there exist $c$ such that $...
- 1,331
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1
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402
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Boundedness of sequences and cardinality
Let $X$ be a set of sequences of real numbers that converge to zero with the property that for any unbounded sequence of real numbers $(y_n)$, there is a sequence $(x_n)$ in $X$ for which the ...
- 121
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1
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467
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Subtracting the weak limit reduces the norm in the limit
Question
Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that
$$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$
...
- 39k
12
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1
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Decomposition of positive definite matrices.
It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum_{j} B_j \otimes C_j
$$
with $B_j$ and $C_j$ positive semidefinite matrices (of ...
12
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4
answers
877
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Can you describe the image of the exponential map $B(H)\to B(H)$?
James Tener asks at the 20-questions seminar:
The exponential map $\exp:B(H)\to B(H)$ is just defined by its Taylor series. Can you describe its image?
- 1,059
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2
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679
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Non-sequential spaces in the wild
TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
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Is there a physical reason that fields in QFT are globally defined?
I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, ...
- 7,952
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Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
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How to think about dual space of a certain space of Lipschitz functions
Consider the following Banach space (for concreteness):
$$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$
where
$$
\bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{...
- 2,478
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2
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Reference for invariance of essential spectrum under relatively compact perturbations
I'm looking for a proof of the following statement:
Let $X$ be a Banach space and $T$ be a closed map on $X$. For any relatively compact map $A$ the essential spectrum of $T$ and $T+A$ are the same.
...
- 191
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1
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838
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A measure theory question
Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
- 1,143
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3
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530
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Making an l_2 distance out of l_1 distance
If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...
- 4,462
12
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1
answer
353
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smooth Luzin theorem
For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...
- 109k
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1
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Can a non-commutative C*-algebra be a minimal operator space?
By an operator space structure on a Banach space $X$ I mean a sequence of norms on spaces $M_n \otimes X$ that satisfies Ruan's axioms.
Among such admissible sequences there is always the smallest ...
- 5,005
12
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2
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547
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Balls in spaces of operators
I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.
Is ...
- 121
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3k
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Infinitesimal generators of stochastic processes
What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator?
More precisely: let $X$ be a measure space ($\sigma$-...
- 22.3k
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1
answer
885
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bornological vector spaces over a non-archimedean field
Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
- 1,180
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1
answer
908
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Equivalence of σ-convex hull and closed convex hull
Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as
$$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
- 123
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2
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811
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Nuclear operators/spaces and transfer operators
While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
- 293