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This question is motivated by an obvious formal analogy between two well-known inequalities: Log-concavity and Brunn-Minkowski inequality Let $\mu(dx) := m(x) dx$ be an absolutely continuous …

Let $A:H\to L^0(S, \mu)$ be a continuous operator from a Hilbert space to the space of (equivalence classes of) measurable functions on a probability measure space $S$ with convergence in measure. Let …

Not in general. It's well-known in Banach space theory that the ideal $c_0$ in $\ell^\infty$ is not complemented (see e.g. Albiac & Kalton). By the Gelfand representation, $\ell^\infty \simeq C(\bet …

For a compact Hausdorff space $K$ the algebra $C(K)$ is $1$-injective iff $K$ is extremally disconnected iff its Boolean algebra of idempotents is complete. So to construct a counterexample we need an …

Let $A$ and $B$ be two bounded symmetric positive operators in Hilbert space, such that $A-B$ is trace class. If needed, $A$ and $B$ may be assumed reasonably "small", let's say, Hilbert-Schmidt. Doe …

Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero b …

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex. We say that $\mu$ admits shifts if f …

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the fo …

Let $X_1,\dots,X_n$ be compact Hausdorff spaces. Let's define the Varopoulos algebra as the projective tensor product: $$V(X_1,\dots,X_n) := C(X_1) \hat{\otimes} \dots \hat{\otimes} C(X_n),$$ i.e. the …

For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator: $M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$ ( …

Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho \in …

Denote the Gaussian random vectors by $X(n)$. Clearly, $\mathrm{tr} \, S(n) = \mathsf{E} \, \Vert X(n) \Vert^2$, so $\mathrm{tr} \, S(n) \to 0$ is certainly sufficient for weak convergence to $0$. An …

There is convergence in some non-locally convex spaces, e.g. $L^p, 0 < p < 1$. More generally, for any concave function $\Psi : \mathbb{R}_+ \to \mathbb{R}_+$, such that $\Psi(0) = 0$ and $\Psi(x) / …

Let $i : X \hookrightarrow Y$ be a dense embedding of complex Hilbert spaces. Let $f : \mathbb{D} \to X$ be a function, such that $i \circ f$ is holomorphic ($\mathbb{D}$ is the open unit disk). I …

Any upper or lower semicontinuous function is continuous almost everywhere in the sense of Baire category (since it is a pointwise limit of a sequence of continuous functions, at least when $\Omega$ i …