# Tagged Questions

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

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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$ where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a non-...
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### 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 ...
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### explicit extention of Lipschitz function (Kirszbraun theorem)

Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H_1$, and $H_2$ is another Hilbert space, and $f : U \to H_2$ is a Lipschitz-continuous map, then $f$ can be extended to a ...
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### Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective? The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...
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 ...