# Tagged Questions

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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### I need to know the most active research topic which depend on Real analysis and functional analysis? [on hold]

I need to know the most current topic in pure math with depend mainly on real analysis and functional analysis and not need a good knowledge in algebra and geomtry ? is delay differntial equations is ...
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Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states. Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\... 1answer 117 views ### Hodge decomposition on open manifold For the open manifold like$X\times \mathbb R$or$X\times \mathbb R^+$, where$X$is a closed manifold. Is there any decomposition like (Hodge Decomposition) of the Differential forms on it. 0answers 92 views ### Gauge Fixing Problem on Cylindrical For Cylindrical$Y\times\mathbb R$, where$Y$is a closed oriented 3-manifold. If it is necessary, we could consider the$b_1(Y)=0$case. Fix a Line bundle$L\to Y\times \mathbb R$and a Hermitian ... 0answers 44 views ### A priori estimates for elliptic operators Suppose$L : L^{m,p}(M)\rightarrow L^p(M)$is some elliptic operator of order$m$, and$(M,g)$is a compact Riemannian manifold. Then it is known that there exists a constant$C$such that we have the ... 1answer 69 views ### Heat kernel upper bounds on a complete Riemannian manifold Let$M$be a complete Riemannian manifold, and$p(t, x, y)$denotes its heat kernel. I am trying to find sufficient conditions for when the following holds: $$p(t, x, y) \leq Ct^{-n/2}, \forall x, y, ... 0answers 49 views ### L^\infty-contractive semigroups Let L^\infty(\mathbb T) be the space of 2\pi-periodic and bounded measurable functions and \mathcal P be a pseudo-differential operator defined on \mathcal D(\mathcal P)\subset L^\infty(\... 1answer 61 views ### Domain of the Stokes operator Let \Omega\subseteq\mathbb R^d be open (d\in\mathbb N) \mathcal D:=C_c^\infty(\Omega)^d and$$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$\mathcal H:=\overline{\mathfrak ... 0answers 41 views ### Questions about the regularity of the solution of the heat equation in a bounded domain [closed] I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose u\in C^2_1(U_T) solves the heat equation in U_T. Then u\in C^\infty(U_T) Here is the statement and ... 1answer 47 views ### About the critical points of quasi-convex functions What do we know about the structure of critical points of quasi-convex functions? I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ... 1answer 81 views ### Reference on Probability theory on functional spaces (in special Hilbert spaces) Currently, I am working on some sort of stochastic optimization problems defined over function spaces. I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")... 2answers 295 views ### Is a C*-algebra with an isomorphic predual a von Neumann algebra? It is well-known that a C*-algebra A is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space X such that A is isometrically ... 1answer 110 views ### Space time Lesbesgue spaces I have a function which lives in f(x,t)∈L^2(0,T;H^{1/2})∩L^\infty(0,T;L^2) for a certain time interval. I also know that \partial_{t} \ f(x,t)∈L^2(0,T;H^{−1}). Can I assure that the function lives ... 1answer 117 views ### Density of smooth functions on Hölder spaces The following result is often cited without reference in the context of PDEs: Let \varOmega \subset\mathbb R^n be a bounded open set with smooth boundary. If 0<\beta<\alpha<1 then C^\... 0answers 28 views ### About a particular definition of a Hessian of a function of tuples of matrices Say I have a function L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R} i.e it takes a tuple of n matrices of different dimensions and computes a number from them. Then I see being defined a ... 0answers 93 views ### Banach spaces complemented in their ultrapowers By the principle of local reflexivity, the second dual X^{**} of a Banach space X is complemented in some ultrapower X^U of X. Even when X is separable, the index set of U cannot be ... 0answers 92 views ### The eigenfunction of modified 1-laplace equation? Let \Omega\subset \mathbb R^2 be open bounded with smooth boundary. It is well known that the laplace equation$$ -\Delta u=0 $$has a set of eigenvalues 0<\lambda_1<\lambda_2\leq\lambda_3<... 0answers 112 views ### Is an bijective analytic map bi-analytic? Suppose that E and F are complex Banach spaces and U\subset E and V\subset F are open subses. f\colon U\to V is analytic f\colon U\to V is bijective Is f bi-analytic? (i.e. is its ... 0answers 49 views ### L^\infty bounds for pseudo-differential equations of parabolic type It is well-known that if the solution of u_t=u_{xx}, with t>0 and x\in\mathbb R, is bounded, then a(t)=\sup_{x\in \mathbb R}u(x,t) is non-increasing, while b(t)=\inf_{x\in \mathbb R}u(x,t)... 0answers 223 views ### Baum Connes Conjecture [closed] I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as it is formulated in topology to the conjecture as it is formulated in functional analysis. I ... 1answer 250 views ### Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum? Let A = \mathcal{C}(X) be a commutative (unital) C*-Algebra. Let Spec(A) denote its Gelfand spectrum$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$Now ... 0answers 124 views ### Norm of projection onto functions of mean zero Let X be a finite set and consider the space \ell^2(X;Y) of functions \zeta:X\to Y, where Y is a fixed Banach space. It decomposes into a direct sum of constant function and its complement \... 0answers 38 views ### References for the Sturm oscillation theorem What is the most general form of the Sturm oscillation theorem? So far I have only seen cases that treat either unbounded intervals or weighted L^2 spaces. I would be especially interested in ... 2answers 370 views ### Extracting subsequences in Banach spaces, along an ultrafilter? There are various principles in Banach space theory that allow one to pass from a given sequence of vectors (x_n), to a subsequence (x_{n_k}) with some desired property. I'm thinking here, in ... 1answer 201 views ### Restriction of irreducible unitary representation to normal subgroup of finite index Let G be a Lie group (or more generally a locally compact group), let N be a closed and normal subgroup of G of finite index. Let H be an infinite dimensional complex Hilbert space, and let \... 1answer 354 views ### Is the L^1-space dual to a Banach space Let (\Omega,\mu) be a measure space. It is well known that for 1<p\leq \infty one has the duality$$L^p=(L^{p*})^*,$$where 1/p+1/p^*=1. Question. Is it known that the Banach space L^1 is ... 0answers 77 views ### If f_j\to f in L^1(\Bbb R^n) then Tf_j\to Tf in L^{1,\infty}(\Bbb R^n) Let's define A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}. So A is dense in L^1(\Bbb R^n). Given then f\in L^1(\Bbb R^n); by density there exists \{f_j\}_j\... 0answers 66 views ### Elliptic regularity on the hypercube Assume$$ Lu=f\quad \text{in } [0,1]^d\\ u=0 \quad\text{ on } \partial[0,1]^d $$for some strongly-elliptic operator L, and f\in H^k$$, k\geq -1$. Do we have $u\in H^{k+2}$? I can only find the ...
I have the question about finding the subspace of concentration of a Gaussian Measure. More precisely: $\textbf{Question:}$ Assume we have a separable Hilbert space $\ell_2$ with Borel $\sigma$-...