# 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.

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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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### 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 ...
1 vote
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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 ...
• 233
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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 ...
• 1,031
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• 9,320
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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-$*$ ...
1 vote
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• 211
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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 ...
• 21.3k
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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 ...
147 views

### 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-...
• 7,257
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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 $(K,\... • 7,257 2 votes 0 answers 122 views ### Differentiability of Norms of$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_{\...
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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 $M\cap N^{\perp} \ne \{0\}$. ...
• 4,300
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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 ? \frac{\delta}{\delta g^{\mu\nu}}\nabla_{\sigma}\Bigr(\alpha(x^{\beta})\,\frac{\nabla^{\sigma}{\phi(x^\beta)}}{\phi(x^\...
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1 vote
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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.
• 1,302
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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 ...
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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 ...