Questions tagged [geometric-functional-analysis]
This tag is for questions relating to "geometric functional analysis", lies at the interface of convex geometry, functional analysis and probability. It has numerous applications ranging from geometry of numbers and random matrices in pure mathematics to geometric tomography and signal processing in engineering and numerical optimization and learning theory in computer science.
43 questions
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Extreme points of a set of bounded increasing functions with a fixed integral.
I am interested in the extreme points of the set of functions 𝑓:[0,1]→[0,1] such that (i) 𝑓 is non-decreasing and (ii) $\int_0^1 f(x) dx = C$ for some $0<C<1$.
Without (ii), I can see that ...
- 17
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76
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Extreme point of the unit ball of $K(X, Y)^*$.
Let $K(X, Y)$ be the collection of all compact linear operators from $X$ to $Y$ and $K(X, Y)^*$ be the dual of $K(X, Y)$. I am interested to know the extreme points of the unit ball of $K(X, Y)^*$. In ...
- 572
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How to show a point is a weak* -weak continuous for the identity map on $X^*$ or on $X^{**}$?
I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak*
-Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
- 572
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55
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Application of Geometry of Banach spaces in modern days or in other studies
I plan to pursue my higher studies in the geometry of Banach spaces. I am curious about the recent era of artificial intelligence, machine learning etc. Is there any field where Banach space geometry ...
- 572
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47
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Does the set of all support functionals at a point $x\in B_X$ contains a functional which is not a convex combination of its extreme functionals?
Let $X$ be a Banach space. $X^*$ denotes the dual of $X$. Let $x\in X$ be non-zero. Let $J(x)$ be the collection of all support functionals at $x$, defined as:
$$J(x)=\{f\in X^*:\|f\|=1, f(x)=\|x\|\}$$...
- 572
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1
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Give a constructive example of a limit point of the convex hull of the unit ball in infinite dimensional Banach space.
I was studying Krein-Milman Theorem in the aspect of infinite dimensional Banach spaces. We know that the closed unit ball $B_{X^*}=\{f\in X^*:\|f\|\leq 1\}$ of dual space $X^*$ of an infinite ...
- 572
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66
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Spanning set of support functionals in dual space
I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries:
Let $X$ be a normed space and $X^*$ be the ...
- 572
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166
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Converse to Bauer's maximum principle
Let $X$ be a Hausdorff real locally convex topological vector space and $K$ be a nonempty compact subset. Bauer's maximum principle states that a convex upper semicontinuous function $f\colon K \to \...
- 546
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131
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Given a rectangle and a equilateral triangle, is it possible to construct a line that bisects the area of the triangle and rectangle?
The pancake theorem guarantees that there is a line that can bisect a rectangle and a triangle simultaneously. The Borsuk Ulam Theorem can be used to prove that theorem. Show that any line through the ...
- 21
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143
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The existence and uniqueness of the curvature of a Yang-Mills connection.
I am reading Jost's Riemannian Geometry and Geometric Analysis (7-Ed) and having a question about the Yang-Mills functional (page 183).
Let $M$ be a compact manifold and $E$ be a metric bundle with ...
- 727
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How to show that total boundedness in $\mathcal{N}$ topology is equivalent to having finite $(\varepsilon,\rho)$-net for each $\rho\in\mathcal{N}$?
Let $X$ be a linear space (over $\mathbb{R}$) with $\mathcal{N}$ a family of semi-norms on $X,$ the $\mathcal{N}$- topology on $X$ is the weakest topology that makes each $\rho\in X$ continuous. Write ...
- 716
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For two commuting contractions $T_1,T_2$ on a Hilbert space $H$ with $T=T_1T_2$ Show that $D_T>D_{T_i}$ for $i=1,2$ where $D_T=(I-T^*T)^{1/2}$.
Let $T_1, T_2$ be commuting contractions on a Hilbert space $H$, that is $\|T_1\|\leq 1$ and $\|T_2\|\leq 1$, $T_1T_2=T_2T_1$. Let $T=T_1T_2$. Clearly $I-T^*T$ and $I-T_i^*T_i$ for $i=1,2$ are ...
- 391
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116
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Stone's Lemma for disjoint convex subsets
Lemma: If A and B are disjoint convex sets in a linear space X, then there are complementary convex sets C containing A and D containing B.
I am genuinely confused by this proof given in the book ...
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66
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Factorization theorem (change of density)
I am starting to take an interest in convex geometry and stumbled on the following theorem due to Pisier, the proof should be in https://link.springer.com/content/pdf/10.1007/BF01450929.pdf, although ...
- 343
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62
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Possible reverse triangle inequality
I'm looking at the convergence (when blowing up the metric) of the spectrum of a self-adjoint operator $P$ that acts on differential forms of a 3-dimensional closed manifold M.
Let $\lambda$ be a ...
- 1,071
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51
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Sobolev space of differential forms [duplicate]
I came across the following definition:
The Sobolev space $W^{k,p}_1(M)$ is the space of differential forms $\alpha\in\Omega^pM$ such that
$$\|\alpha\|^2_0=\int_M\alpha\wedge \star\alpha<\infty \...
- 1,071
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2
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343
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Geometric implication of the Sobolev embedding
It is stated in section 10 of this paper that the usual Sobolev embedding $$W^{1,1}(\mathbb{R}^n) \subset L^{n/(n-1)}(\mathbb{R}^n)$$ can be interpreted in geometrical terms as an isoperimetric ...
- 2,882
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1
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140
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Variance of Geometric Distribution in terms of successes only [duplicate]
$$ Prove: var(X) = \frac{1-p}{p^2} $$
I solved for $$E[X^2]-E[X]^2$$
I did the following for $E(X)$
$$ E(X) = (\frac{1}{p}) $$
I did the following for $E(X^2)$
$$E(X^2)=\sum^\infty_n n^2p(1-p)^n$$
$$E(...
- 1
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1
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75
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Euler Lagrange Equation and Besse Conjecture
Many paper said that Einstein Hilbert functional $E(g)$ defined as follows $$E(g) = \int_{M} R_{g}dM_{g}$$
If it restricted on unit volume. The Euler Lagrange can be writen as $$Ric - \frac{R}{n} = ...
- 133
1
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1
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54
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Prove the existence of $\Psi\in(\ell_\infty(G))^\ast$ satisfying the amenability conditions
Let $G$ be a finitely generated group satisfying the Folner condition and let $S\subset G$ be a finite generating set of $G$. Denote by $\rho=\{\rho_g\}_{g\in G}$ the right translation action of $G$ ...
4
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1
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65
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For discrete group $G$ and $H\leq G$. Show that $G$ also satisfies the Folner condition if $H$ satisfies it and $[G:H]<\infty$. [closed]
A finitely generated group $G=\langle S \rangle$ is said to have the Folner condition if $\forall \varepsilon>0$, there exists a finite subset $F\subset G$ such that
$$\#((S\cup S^{-1})F\setminus F)...
2
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1
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283
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Extreme Points of Disc Algebra
I know that the extreme points of the closed unit ball of $\mathcal{H}^\infty(\mathbb{D})$, the space of all bounded holomorphic functions on the unit disc are the functions $f\in\mathcal{H}^\infty(\...
- 21
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1
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554
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What is Banach-Dieudonne theorem?
Let $X$ be a separable Banach space and $X^*$ be its dual and $w^*$ be weak$^*$ topology on $X^*$. Let $B(X^*)$ denotes closed unit ball of $X^*.$
I'm reading a research paper in which we want to ...
- 75
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100
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Convergence to an $\ell_p$ ball, of Steiner symmetrization of compact convex subsets of $\mathbb R^n$
Context. I'm working on a problem, and it seems Steiner symmetrization might just be the golden trick. But first, I must make sure the process will converge to an $\ell_p$ ball...
Fix $p \in [1,\...
- 9,635
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73
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Shorter proof that the fiber of an extreme point contains an extreme point
I think I have a proof of the following result:
Let $V$ be a separable real Banach space. Let $M \subset V^*$ be a nonempty convex subset of the unit ball in $V^*$ which is closed in the weak-$*$ ...
- 918
1
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1
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159
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Sharp radius for univalent convex functions
Context
Koebe 1/4 Theorem states the following:
Theorem.- Let $f \in \mathcal{S}$, that is, the set of univalent (analytic and injective) with $f(0)=0$ and $f’(0)=1$ functions from $\mathbb{D}=\...
- 469
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38
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refering something as "non-linear" when there is no underlying linear structure
Can I talk about a non-linear shape functional. I understand a shape functional $J$ as some mapping that takes a shape and returns a real (or complex) value.
I would like to talk about a non-linear ...
- 513
2
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1
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348
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Applications of Sard's Theorem.
I am writing my Bachelors Thesis about Sard's Theorem and I was asking myself if there are any good applications of it or the direct consequences (Whitneys Embedding and Morse functions) in physics or ...
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1
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52
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Proving three convex surfaces intersect using 2 Dimensional intermediate value theorem
I have three convex functions $f_1(x,y), f_2(x,y),$ and $f_3(x,y)$. I know that $f_1(x,y)$ is non decreasing on both $x$ and $y$, and $f_3(x,y)$ is non increasing on both $x$ and $y$. $f_2(x,y)$ is ...
- 2,577
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96
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Homeomorphism between a piece of sphere and a convex set on a Banach space.
Let $(X,|\cdot|)$ be a real Banach space and $K\subseteq X$ a closed cone, that is, $K$ is a nonempty closed subset of $X$ satisfying: (a) $0\in K$; (b) $x+y\in K$ whenever $x,y\in K$; (c) $tx\in K$ ...
- 4,398
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1
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181
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Proving that the norm of a linear and surjective operator is 1 (Step in Figiel's Theorem)
In the book Geometric Nonlinear Functional Analysis by Y.Benyamini and J.Lindenstrauss, in Theorem 14.2 (page 343), due to Figiel, the authors prove that a certain operator has norm $1$.
This ...
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Given a "composite" norm, what polygon describes its unit ball?
When answering this question about finding the open unit ball $\mathscr{B} := \{ x \in \mathbb{R}^2: \| x \| < 1\}$ of the "composite" norm
$$
\| \cdot \|:
\mathbb{R}^2 \to \mathbb{R}, \
(...
- 4,878
2
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2
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872
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Hahn Banach Theorem implying existence of a nonzero linear functional taking 0 in a linear subspace
I am reading this paper. In the proof of theorem 1, it is stated
By the Hahn-Banach theorem, there is a bounded linear functional on $C(I_n)$, call it $L$, with the property that $L\ne 0$ but $L(R)...
- 211
2
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1
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103
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Prove that $\forall t \in \mathbb R$ the set $f^{-1}(\{t\})$ is a hyperplane of $X$
Exercise :
Let $X$ be a vector space and $f:X \to \mathbb R$ be a linear functional. Show that for all $t \in \mathbb R$, the set $f^{-1}(\{t\})$ is a hyperplane of $X$.
Attempt :
I have proved a ...
- 21.5k
2
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237
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Distance between two functions in term of a third function
I am wondering if it is possible to define a distance between two real-valued functions that express the concept of "the space between them that can be filled by a third function", where the third ...
- 233
5
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1
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173
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Definition of the Berkovich spectrum
I am trying to read these notes:
http://www-personal.umich.edu/~takumim/Berkovich.pdf
Regarding the Berkovich spectrum. In definition [2.24] it says that the spectrum is the set of bounded (non-...
- 8,064
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1
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331
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Realizing the Berkovich affine line as a union of Berkovich spectrums
I am trying to understand what is the relation of the affine Berkovich space to the Berkovich space on an appropriate polynomial ring. A more exact version of the question is as follows:
Let $(K,\...
- 8,064
2
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0
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126
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Differentiability of Norms of $l_{\infty}$
In the book Fabian and others I saw exercise:
"Let $\|$.$\|_{\infty}$ denote the canonical of $l_{\infty}$ and set $p(x) = \limsup |x_i|$. Define $\||x\|| = \|x\|_{\infty} + p(x)$ for $x \in l_{\...
- 21
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About non-separable Hilbert spaces
On Reed & Simon, vol 2, chapter X, problem 4, it is asked:
Let $M$ and $N$ be closed subspaces of a separable Hilbert space. If $\dim M > \dim N$, prove that $M\cap N^{\perp} \ne \{0\}$.
...
- 4,398
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1
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How to vary a second order function with respect to the metric tensor?
Can anybody help me to prove this relation, how is it is valid ?
\begin{equation}
\frac{\delta}{\delta g^{\mu\nu}}\nabla_{\sigma}\Bigr(\alpha(x^{\beta})\,\frac{\nabla^{\sigma}{\phi(x^\beta)}}{\phi(x^\...
- 489
1
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0
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252
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Bound degrees of sparse random graphs
I might be wrong but I think this problem (Exercise 2.4.2) means $d=o(\log n)$? If so, can anyone give a hint instead of telling the answer.
- 1,332
3
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1
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278
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A good resource on the Radon-Nikodym Property in reflexive Banach Spaces?
I'm looking for a good resource that builds the theory of the Radon-Nikodym Property. I'm not particularly interested in the measure-theoretic characterisation; I'd like the geometry of Banach Spaces ...
- 53.2k
7
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2
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753
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Concentration of norm of projection onto a subspace
Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of $\mathbb{R}^n$ of dimension $k$ and let $P_V(x)$ be the orthogonal projection of ...
- 26.6k