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Here is a nice exposition due to Maurey: http://www.numdam.org/numdam-bin/fitem?id=SAF_1972-1973____A10_0 http://www.numdam.org/numdam-bin/fitem?id=SAF_1972-1973____A11_0

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 …

It's not sequential because its closed subspace $M[0,1] = (C[0,1])^\ast$ is not sequential. Here is an example of a set $A \subset M[0,1]$ that is sequentially $\tau_v$-closed but not $\tau_v$-cl …

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 …

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 …

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 …

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 …

The projection from the space $c$ of convergent sequences to the $1$-dimensional subspace of constant sequences with kernel $c_0$. Any projection $c \to c_0$ has norm at least $2$ (cf. exercises to ch …

I would guess "the" natural topology on this space should be that of uniform convergence on compact subsets. Since a function on $X$ is continuous iff it its restrictions on compact subsets are contin …

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 $A$ 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 …

The problem reduces to the case of smooth functions with compact support, since they are norm dense in $C_0$. Now let $f$ be a smooth function with compact support. Then $[T,f]$ is an integral operat …

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 …

Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties? The distribution of $X$ is log-concave, i.e. for every $n$ the joint distr …