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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).

Functional analysis is 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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Are there any results giving rates of weak convergence of orthonormal sequences to zero?

Motivated by this question, I am interested in the following: Let $\,(x_n^1)_{n \geq 1}$, $(x_n^2)_{n \geq 1}$, $(x_n^3)_{n \geq 1}$, $\,\ldots\, $ be an orthonormal sequence of square-integrable real sequences $(x_n^i)_{n \geq 1} \in…
Julian Newman
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For what value of $a$, $f(x)-\tan^{-1} (x^2+x+a)$ is surjective from $\mathbb{R}$ to $\left( 0, \frac{\pi}{2} \right)$

$\tan y=(x^2+x+a)$ where $\tan y$ is positive in $\left(0, \dfrac{\pi}{2} \right)$ so the quadratic equation is also positive. Now how to get value of $a$?
Partha sarathy
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Problem about Hahn-Banach Theorem

Could someone help me with the following problem? Let be $X$ a real normed space , $f,g\in{X^*}$ ($\|f\|=\|g\|=1$) and $0
mathlife
  • 649
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Application of Baire category theorem on functions

I have to solve the following task by using the Baire category theorem: Let be $f: [0,\infty] \rightarrow \mathbb{R}$ a continous function such that $$\forall t \geq 0: \lim_{n \rightarrow \infty} f(nt) = 0 $$ I have to show that $f$ satifies…
Gustav.G
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Is $\mathbb{R}$ a subspace of the Euclidean Plane $\mathbb{R}^2$

It said that $\mathbb{R}$ a subspace of the Euclidean Plane $\mathbb{R}^2$. But $\mathbb{R}$ is not even a subset of $\mathbb{R}^2$. How can it be possible? Thank you for helping.
MH Yip
  • 199
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weak convergence of nth unit vector in $\ell^p (\mathbb N)$

I would like to know, Why is it true that $e_n$, the nth unit vector in $\ell^p(\mathbb N)$ converges weakly to $0$. $1 < p < \infty$ According to Mazur's lemma, which says that $y_n$ is convex combination of $e_n$ converges to $0$ in norm…
Theorem
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Is it true that $\|Ay\|_S\leq \|A\|_S\|y\|_S,\;\forall y\in E\;?$

Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$. For $S\in \mathcal{L}(E)^+$, we consider the following…
Student
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On the second geometric form of Hahn-Banach´s theorem (Brezis, theorem 1.7) I dont know if im understanding this inequality correctly

f is a functional. $f(x-y)\leq f(rz),$ $\forall x\in A,$ $\forall y\in B$, $\forall z\in B(0,1)$ Then $f(x-y)\leq -r ||f||,$ $\forall x\in A,$ $\forall y\in B$ The problem is with the sign of the r is it because the z is on the B(0,1) so…
Felipe Trajano
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Orthogonal space, functional analysis

Is $E$ a normed banach space and is $F\subseteq E$ a closed, complemented subspace of $E$, then is $F^\bot=\{f\in E'\colon f_{|F}= 0\}$ a closed, complemented subspace of $E'$. I do not know if I translate "complemented" right. So here is the…
Cornman
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Generalisation of the norm of bounded linear operators

Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$. Let $M\in \mathcal{L}(E)^+$ (i.e. $M^*=M$ and $\langle…
Student
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Show that a such set of Hahn-Banach extensions is infinite

Let $M$ be the subspace of $C([-1,1])$ consisting of all even functions. Let $\psi:M\to \mathbb{R}$ be the linear functional given by $\psi(f)=\int_{-1}^{1}f(t)dt$. Show that there are infinitely many bounded linear extensions $F$ of $\psi$ to…
Hopeless
  • 906
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Sequence of convex combinations

Let $(e_n)_{n\in \mathbb{N}}$ be unit vectors in a normed space X. I search for a sequence $(y_k)_{k\in \mathbb{N}}$ of convex combinations of the $(e_n)_{n\in\mathbb{N}}$, which converge strongly to $0$. Can someone give me an example? I already…
Tobi92sr
  • 1,661
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Want show $\sup_n |T_n(x)|=\infty$ for some $x\in \ell_p(\mathbb N)$

Let $p\in (1,\infty)$. For each $n\geq 1$, let $T_n:\ell_p(\mathbb N)\to \mathbb C$ be the continuous linear functional given by $T_n(x)=\sum_{i=n}^{2n}x_i$, where $x=(x_k)_{k\geq 1}\in \ell_p(\mathbb N)$. The claims to be proven are: i) $\sup_n…
Hopeless
  • 906
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1 answer

Showing equivalence between norm and norm defined by a Schauder basis

Let $X$ be a Banach Space with norm $\|\cdot\|$. Let $\{x_n\}_{n\in\mathbb{N}}$ be a Schauder basis of $X$, let $x=\sum c_n x_n$, define a new norm $$\{\{x\}\}=\sup_{N \in \mathbb{N}} \left\|\sum_{n=1}^N c_n x_n\right\|$$ I need to show that those…
Amit
  • 652
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Duality between $\ell^p$ and $\ell^q$

I have the following problem. I need to show that for a bounded sequence (of sequences) $\{a_n\}\subset \ell^q$, such that $\lim_{n\to\infty} a^n_i = 0$, i.e. the sequence converges coordinate-wise to zero, the following holds for all…
The Brainlet Exterminator
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