# Questions tagged [fa.functional-analysis]

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

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### Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces

The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is, $[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$. Now I ...
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### Relationship between Beilinson’s resolution of the diagonal and functional analysis

I have been wondering for long enough to embarrass myself on here by asking: is there a reason why Beilinson’s resolution of the diagonal “Coherent Sheaves on Pn and Some Problems of Linear Algebra” ...
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### States "absorbed" by a Haar idempotent on a compact quantum group

Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when $$a\bullet b=b=b\bullet a?$$ Can we say that $b$ absorbs $a$? Can we say ...
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### Finding set of best approximations from a point in $c_0$ to its subspace

Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
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### Reference for Schwartz kernel theorem on vector bundles

In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
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### Sufficient condition for two norms to be equal

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$. On $\mathcal{L}(E)^2$, we have two equivalent norms: \begin{eqnarray*} N_1(A,B) &=&\sup\left\...
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### How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]

I noticed that in the paper  this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...
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### Intersection of Hilbert spaces with Schauder basis plus a fixed singleton

Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$. Let $U \subset H$ be a Hilbert subspace of $H$. Let $\{u_n\}_{n \in \mathbb{N}}$ be a Schauder basis for $U$. Let $h \in H$...
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### Could we characterize elements in the second dual by the character space?

Let $A$ and $B$ be two semisimple commutative Banach algebras. Assume that $A\mathbin{\tilde\otimes} B$ is a Banach algebra obtained by completing $A\otimes B$ with respect to a cross norm not ...
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### Rigorous proof of the pentagon identity

I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra. For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...
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