# Questions tagged [fa.functional-analysis]

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

5,349 questions
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### Can we have Levy area for N dimensional process?

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there a equivalent area for N dimensional Brownian motion, if so what ...
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### On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$

Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
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### Connectness of $K$ and the existence of non-trivial $M$-summands on $C_0(K)$

For a locally compact Hausdorff space $K$, we denote by $C_0(K)$ the Banach space of continuous functions from $K$ to $\mathbb R$ which vanish at infinity equipped with the usual supremum norm. For ...
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### Eigenfunctions of square root of Laplacian in an arbitrary Riemannian manifold?

Please forgive me for my inability to pose a mathematics question properly. In one dimension the eigenfunctions of Laplacian (simply double derivative) are also eigenfunctions of its square root (...
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### Maps which are both completely positive and positive

Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
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### Predictable Process Question (Da Prato & Zabcyzk 2014)

I've been looking at the 2014 edition of Da Prato & Zabcyzk and the sections on predictable processes. In particular, in their Proposition 3.7 (ii), they assert that if $\Phi$ is adapted and ...
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When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that $$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ... 1answer 266 views ### Does the Banach space ( \ell ^2 \oplus \ell ^2 ) have F.P.P? The space ( \ell^2 ,\lVert \cdot \rVert _2 ) is a Hilbert space. The space X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty ) is a Banach space. Does X have fixed point property? (For any ... 0answers 80 views ### Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates Let M a two-dimensional compact manifold and f:M\to M a diffeomorphism C^r, r\geq 2 and f(x,y)=(mx,\lambda y) where m:M\to \mathbb{R} and \lambda:M\to \mathbb{R} ,\lambda<1<m. ... 1answer 196 views ### Fundamental group and group measure space construction Let N be a type {\rm II} factor, with trace \tau. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
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Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...
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### Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example. Consider the interval $[-\pi,\pi]$ let's say. ...
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### Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: $$C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).$$ what condition should be put ...