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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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Operator norm convergence and pointwise convergence

I just want to make sure I am understanding this correctly. I want to show that operator norm convergence implies pointwise convergence. Suppose we are looking at the space $L(X,Y)$ i.e the space of all linear operator from X to Y. Suppose that $S_m…
user329017
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1 answer

Uniqueness of annihilator subspace in infinite dimensional normed space

Let $Y_1$ and $Y_2$ be closed subspaces of a normed space $X$ (which may be infinite dimensional). How can I show that $Y_1$ and $Y_2$ must have different annihilators or else be the same subspace? The annihilator $Y^0$ of a subspace $Y \subset X$…
Jessica Hansen
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If $\{x^*(x_\lambda); \lambda\in\Lambda\}$ is bounded for each $x^*\in X^*$, then the set $\{x_\lambda; \lambda\in\Lambda\}$ is bounded

Problem $X$ is a Banach space. $\{x_\lambda\}_{\lambda\in\Lambda} \in X.$ $X^*$ is the dual space. Show the following. If $\forall x^*\in X^*, |x^*(x_\lambda)|\le M \Rightarrow \{\|x_\lambda\|\}_{\lambda\in \Lambda}$ is bounded. ($M$ is a…
Arbitrary
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Prove that finite dimentional subspaces of normed vector spaces are topologically complemented.

So I'm trying to solve the following exercise: If $X$ is a normed space and $Y$ a finite dimensional subspace. Show that $Y$ is complemented in $X.$ I know there's a proof of it using the Hahn-Banach Theorem, however I'm trying to prove it without…
H_Hassan
  • 807
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Joint euclidean norm of operators

In this question $E$ stands for a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{L}(E)$ the algebra of all bounded linear operators from $E$…
Student
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Existence of only one bounded linear functional.

Given $X$ a linear normed space over $K$ . $I$ be arbitrary indexing set , $\{f_\alpha: \alpha \in I\}\subset X$ and a family $\{c_\alpha: \alpha\in I\} \subset K$, I want to know that there exists exactly one bounded linear functional $f'\in X'$…
Theorem
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Norm of linear functional; can we take supremum over dense subset?

If $V \subset H$ are Hilbert spaces and $V$ is dense in $H$, is it true that for $f \in H^*$, $$\lVert f \rVert_{H^*} = \sup_{v \in V} \frac{|f(v)|}{\lVert v \rVert_V}?$$ So I mean can we just take the supremum in the definition of the norm of $f$…
maximumtag
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How to show a collection is a local subbasis of the weak topology?

Let $X$ be a normed $\mathbb K$-linear space and for all $x \in X$ , $f \in X^*$, $\epsilon >0$ Let us define $$U_x(f,\epsilon)=\{y \in X : |f(y)-f(x)|<\epsilon\}\\ =x+f^{-1}(B(0,\epsilon))$$ Then for each $x \in X,$ $\{U_x(f,\epsilon)|f \in X^*,…
Mini_me
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Example for $A,B$ non-empty, bounded, and closed, $A\cap B = \emptyset$, but $\operatorname{dist}(A,B) = 0$

Is there such an example in a Banach space? Note that this is not possible in a finite-dimensional space. In what I have here, $A$ and $B$ are the closures of relatively open sets in a bounded and closed set.
amsmath
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$T(B_E)$ is closed for $T$ a bounded linear map

Let $T:E\rightarrow F$ be a bounded linear map for E and F Banach spaces, and E reflexive. Let $B_E$ be the unitary closed ball in $E$. How would you argue that $T(B_E)$ is closed?
Rojas Azules
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Every finite dimensional topological separated vector space is isomorphic and homeomorphic to $\mathbb{K}^{n}$.

I have a question with the following theorem, Every finite dimensional topological separated vector space, over the $\mathbb{K}$(=$\mathbb{R}$ o $\mathbb{C}$), is isomorphic and homeomorphic to $\mathbb{K}^{n}$. In class, we saw the prove of this…
user470833
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3 answers

Compact Operator on banach espace

Hi I just dont know if this proposition is true, I think it is but I dont know how to start: Let $X$ be a Banach space of infinite dimension, if $T \in B(x)$ and there is an $N$ such that $T^N=I$ then T is not compact. I clearly know that $T^N$ is…
Porufes
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Calculate the operator norm on L2

I have to find the operator norm of $A$ as an operator on the Hilbert space $L^2([0,\pi])$, where $A$ is defined as $$A(f)(x) = \int_0^\pi \sin(x-y) f(y)dy,\text{ where } 0 \le x \le \pi.$$ So the idea I have from lecture notes is to use the…
Supersalt
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Compact linear operators composition on Banach space implies that one of them is compact?

Hi i just have some question regarding this problem , I cannot find a counter example and neither a proof: Let $X$ be a Banach space of infinite dimension, and $S,T\in B(X)$ (the set of bounded linear operator from $X$ to $X$) is it true that: if…
Maria
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How can we show that the following is a norm

Let $X=K^3$. For $x=(x(1),x(2),x(3))\in X$, let $||x||=[(|x(1)|^2+|x(2)|^2)^\frac{3}{2}+|x(3)|^3]^\frac{1}{3}$. Then $||.||$ is a norm on $K^3$
user44948
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