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      Questions tagged [fa.functional-analysis]

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

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      0answers
      24 views

      Approximation of functions by tensor products

      Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...
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      votes
      0answers
      47 views

      Minimum intersection of two $\ell_p$-norm balls separated by a fixed $\ell_1$-distance

      Given two $\ell_p$ norm balls in $\mathbb{R}^d$, $$B_a(x) = \{ z\in \mathbb{R}^d: \|x - z\|_p \leq a \}$$ and $$B_b(x + \delta) = \{ z \in \mathbb{R}^d: \|x + \delta - z\|_p \leq b \}\hbox{ with }x \...
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      votes
      0answers
      44 views

      Approximation of functions in $L^p(R^d;L^\infty)$

      Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that $$...
      3
      votes
      1answer
      45 views

      Question on relation between a parabolic sobolev space and a sobolev bochner space

      For parabolic sobolev spaces I follow the following definition: According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$ ...
      3
      votes
      0answers
      64 views

      Tensor product of compact operators on Banach modules

      Let $A$ and $B$ be Banach algebras. Consider a right Banach $A$-module, $E$, and a right Banach $B$-module, $F$, as well as a Banach algebra morphism $\pi\colon A\to\mathcal L_B(F)$ into the bounded $...
      3
      votes
      1answer
      256 views

      Where to find the proof of this property?

      I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...
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      votes
      0answers
      47 views

      What is the relationship (mapping) from a reciprocal function 1/r to a exponential function exp(-r)? [on hold]

      The mathematical problem: Consider the mapping from $r$ to $u$. for large $r$ the theory suggests a formulation like $u=a_1 e^{-a_2 r}$, which means that the function decay exponentially. for ...
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      vote
      0answers
      90 views

      Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$

      Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$: $$ V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}. $$ ...
      3
      votes
      1answer
      61 views

      Mapping inclusion theorem for the numerical range

      We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$. Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
      4
      votes
      1answer
      81 views

      Complemented subspaces of Lorentz sequence spaces?

      Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight. Is there very much known ...
      3
      votes
      0answers
      63 views

      Dual Lorentz spaces

      MO seems the perfect place to ask for the following question. Denote the Lorentz spaces on an arbitrary measure space $(E,\mu)$ by $L^{p,q}=L^{p,q}(E,\mu)$, and by $p'$ the conjugate index of $p$. ...
      5
      votes
      1answer
      189 views

      Quasinilpotent , non-compact operators

      If $X$ is a separable Banach space and $(\epsilon_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon_n$ for all $n$? I suspect the answer ...
      3
      votes
      3answers
      139 views

      Existence of solution to linear fractional equation

      We consider the equation $$?\sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$ where $\lambda_j>0$ and $x_j$ are real distinct numbers. I want to show that if $\lambda_k$ is small compared to the ...
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      votes
      0answers
      66 views

      Duality mapping of a the space of continuous functions? [on hold]

      The duality map $J$ from a Banach space $Y$ to its dual $Y^{*}$ is the multi-valued operator defined by: $J(y)=\{\phi\in Y^{*}:\, \left< y,\phi\right >=\Vert y\Vert^{2}=\Vert \phi \Vert^{2}\},...
      4
      votes
      0answers
      96 views

      Approximation of a compactly supported function by Gaussians

      Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...

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