Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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Existence of a postive measurable set such that $T^{-k}(E)\cap E=\emptyset$ for a particular $k\ge 1.$

Let $(X,\mathcal B,\mu)$ be a probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left(\{x\in X: T^n(x)=x\}\right)=0$ for every $n\ge 1.$ Let $A\in \mathcal B$ ...
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Some questions about integration and operator theory.

As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
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convergent subsequence in $S$ versus in $\overline{S}$

I am reading equivalent definition of a precompact set $S\subset H$, where $H$ is a Banach space. We have $S\subset H$ is precompact iff every sequence in $S$ has a convergent subsequence. But shouldn'...
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Adjoint of sum of two operators

Let $A$ be self-adjoint and $B$ symmetric (which means densely defined for me as well) with $A$-bound less than $1$. Does this imply that $(A+iB)^*=A-iB$ ?
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Prove that this is finite set

Let $(M,g)$ be a Riemannian manifold. Let $\pi_1(M)$ be the fundamental group of $M$. Let $c\in \pi_1(M)$, define $L([c])= inf \lbrace L(\gamma) , \gamma \in [c]\rbrace$. prove that the inf is ...
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Exercise 4.6 - Topics in Banach space theory (Albiac, Kalton)

I'm trying to solve Exercise 4.6 from Albiac and Kalton's book "Topics in Banach space theory". The exercise is as follows: (a) If $K$ is extremally disconnected, show that for every ...
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Bounded linear functional with specific values at linearly independent vectors

Let $(E,\|\cdot\|)$ be a normed vector space. Let $\operatorname{dim}(E) = n < \infty$. Given $\{x_1, \dots, x_n\}$ linearly independent vectors and $a_1, \dots, a_n$ scalars we can always ...
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Computing the spectrum of the operator $A(f)(t)=tf(t), A:L^1([0,1])\to L^1([0,1])$

Let $A$ be the bounded linear operator from the Banach space $L^1([0,1])$ to $L^1([0,1])$ defined as $A(f)(t) = tf(t)$. I've come to learn that then $A$'s spectrum should be $[0,1]$, but I'm a bit ...
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Richard Feynman's Definition of Half Derivative [duplicate]

I read from Leonard Mlodinow's Feynman's Rainbow that Feynman has defined the concept of a half-derivative, an operation $g$ on a function $f$ such that $$g(g(f))=f'.$$ I wonder if it is possible to ...
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Application of Banach-Steinhaus (Uniform Boundness Principle)

Let $E, F$ Banach spaces and $W \subset \mathcal{L}(E,F)$ . If for all $(x,f) \in E \times F' : \sup_{T \in W} |f(Tx)| < \infty$, then $\sup_{T \in W} \lVert T \rVert < \infty$ . I am trying ...
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Ries representation VS nested hilbert spaces [duplicate]

Consider 2 hilbert spaces V ↪H , V continus dens included in H ,V≠H and the duals H* and V* . As far as the proofs I understand why we sey V↪H=H* ↪V*, but I fail to understand why this example does ...
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If $h_k\to h$ in $L^2(\mathbb{R}^n$ then $(h_k\ast \varphi_{\varepsilon_k})\to h$ in$L^2(\mathbb{R}^n)$

Let $(h_k)_{k\in \mathbb{R}^n}\subset L^2(\mathbb{R}^n)$ converge to some $h$ in $L^2(\mathbb{R}^n)$ and $(\varphi_{\varepsilon_ k})_{k\in \mathbb{N}}$ be the standard mollification sequence. I want ...
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Why is that $D_n^ku(x',0)=f_k(x')$? (From H\"{o}rmander‘s Book "Linear Partial Differential Operators" (1969) P.55 Theorem 2.5.7)

Here's part of the proof: However, based on my calculations, $D_n^k\hat{u_n}(\xi',0)=\hat{f_k}(\xi')i^k$, rather than $D_n^k\hat{u_n}(\xi',0)=\hat{f_k}(\xi')$. How come $D_n^ku(x',0)=f_k(x')$?
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When is the image of a linear operator closed?

Let $X$, $Y$ be Banach spaces. Let $T \colon X \to Y$ be a bounded linear operator. Under what circumstances is the image of $T$ closed in $Y$ (except finite-dimensional image). In particular, I ...
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Show a transformation is Conformal

The question I've been given is as follows A linear map $T:X→Y$ is said to be conformal when it preserves orthogonality; $$∀x,{\tilde x}∈X,〈x,{\tilde x}〉= 0⇐⇒ 〈Tx,T{\tilde x}〉= 0.$$ Show that this is ...
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There exists a finitely additive measure on $P (\mathbb R)$ such that $\mu([0,1]) = 1$ and it is a translational invariant.

I have the following question: Show that there exists a finitely additive measure $\mu$ on $P (\mathbb R)$ such that $\mu([0,1]) = 1$ and $\mu(A+x) = \mu(A)$ for all $A \subseteq \mathbb R$ and for ...
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Convolution between $f$ and $g$, with $g$ being in the Schwartz class. Does it follow that $f \ast g \in C^\infty$?

Usually, the convolution between two functions $f,g$ defined on $\mathbb R^n$ is given by $$(f \ast g)(x) = \int_{\mathbb R^n} f(x-y)g(y) \, dy.$$ Right now I am wondering about a specific property ...
Let $(H, \langle \cdot, \cdot \rangle_H)$ and $(U,\langle \cdot, \cdot \rangle_U )$ be Hilbert Spaces such that $H$ embeds into $U$. Let $M$ be a closed subspace of $U$, and define $\mathcal{P}$ to be ...
Integration by parts with $|\nabla|:=\sqrt{-\Delta}$
What conditions on $f,g$ do I need to justify the integration by parts $$\int f|\nabla|g\,dx=\int(|\nabla|f)g\,dx.$$From $|\nabla|:=\sqrt{-\Delta}$ we have formally that $|\nabla|$ is a self-adjoint ...