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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 .

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Some functional analysis problem

In my undergraduate functional analysis course, there is this $1$ problem that I am stuck on, and it goes: For $n = 1, 2, 3,...,$ let the functions $f_n : [0, 1] → [0, 1]$ satisfy $|f_n(x)−f_n(y)|≤|x−y|$ whenever $|x − y| ≥ \frac{1}{n}.$ Prove…
Aurora Borealis
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Is image of ball of finite rank linear operator compact?

Let $X$ be a complex Banach space and $A:X\to \mathbb{C}^{n}$ be a continuous linear map. If $B_{X} = \{x\in X\,:\, ||x||\leq 1\}$ is a closed unit ball in $X$, is it true that $A(B_{X})$ is compact in $\mathbb{C}^{n}$? Since every finite rank…
Seewoo Lee
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Weak convergence implies strong convergence

Assume that the sequence $(A_n)$ of bounded linear operators on a Hilbert space $H$ converges weakly to an operator $A$. Assume also that $\|A_nx\|\to \|Ax\|$ for all $x\in H$. Prove that $(A_n)$ converges strongly to $A$, i.e. $A_n x\to Ax$…
RhythmInk
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Four Pillars of Functional Analysis

I have come across to a statement in many Functional Analysis books saying that "Hahn Banach theorem, Uniform Boundedness Principle, Open mapping theorem and Closed graph theorem are the four pillars of Functional Analysis" I don't exactly know why…
Devendra Singh Rana
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Puzzle on the proof of that test function space is not metrizable

Rudin argues in "functional analysis", the test function space $D(\Omega)$ on nonempty open set $\Omega$ is not metrizable. Let $D_K$ be the subspace of $D(\Omega)$ consisting of functions with support contained in compact $K$. Rudin said that "it…
stephenkk
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Prove that $AB=BA$

Let $E$ be a complex Hilbert space Let $A,B\in \mathcal{L}(E)^+$. Assume that there exists $z\in \mathbb{C}^*$ such that $AB=zBA$. Why $$AB=BA\;?$$
Student
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Weak compactness of Sobolev spaces

I am trying to understand a proof in Evan's book "Partial Differential Equations". We have a sequence $(u_n)_{n\in\mathbb{N}}$ in $L^q(U)$ where $U$ is a bounded open set of $\mathbb{R}^m$. We know that $\sup_n||u_n||_{L^q(U)}<+\infty$ and…
Nicolas
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Numerical radius of a pair of operators in Hilbert spaces

Let $(C,D)$ be a pair of bounded linear operators on a complex Hilbert space $E$. The Euclidean operator radius is defined by $$w_e(C,D)=\displaystyle\sup_{\|x\|=1}\left(|\langle Cx,x \rangle|^2+|\langle Dx,x \rangle|^2\right)^{1/2}.$$ Moreover, the…
Student
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$(X \oplus_p Y)^*$ isometric to $(X^*\oplus_q Y^*)$

Let $X,Y$ be Banach spaces. For $1 < p < \infty$, define a norm on $X \oplus Y$ by $\|(x,y)\|_p=(\|x\|_X^p+\|y\|_Y^p)^{1/p}$. Homework asks to prove that: $(X \oplus_p Y)^*$ is isometric to $(X^*\oplus_q Y^*)$ ($^*$ denotes dual). Ofcourse, my…
Gils
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Some functional calculus of commuting positive operators

Let $A,B$ be two positive operators on a complex Hilbert space. We know that we can define $A^a$ for any $a\geq0$. If $A$ commutes with $B$, then do we have $(AB)^a=A^aB^a$? I believe this is correct but I am not too sure whether my proof is correct…
Math
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Metrizability of weak topology on separable Hilbert space

The weak* topology on the dual of a separable space is metrizable. On a Hilbert space, the weak topology and the weak* topology coincide, and the dual of the Hilbert space is itself. Thus, on a separable Hilbert space, the weak topology is…
keej
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How to show a space is not reflexive?

I am studying functional analysis and I was asked to prove that $c_0$ is not reflexive. The point is I have no idea how to prove this. I don't even know how to show that a space is reflexive. What must I do on the practice? I think it is hard to…
L.F. Cavenaghi
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Orthonormal basis of $L^2(0,2\pi)$

I learned that the functions $e_n = \dfrac{e^{inx}}{\sqrt{2 \pi}}$ are orthonormal basis of $L^2(\mathbb{T})$, the set of square integerable functions on a torus. But then I saw that the same functions are also the orthonormal basis of the space…
nan
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Absolute convergence implies convergence in complete spaces

Let $V$ be a normed space with norm $\|\|$. $\sum_{n=1}^\infty a_n$ is an absolutely convergent series if $\sum_{n=1}^\infty \|a_n\|$ converges. Could you please explain why in complete normed spaces the absolute convergence implies convergence, but…
Konstantin
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Every proper subspace of a normed linear space is not open

Possible Duplicate: Interior of a Subspace How do we show that any proper subspace of a normed linear space is not open. I know that for nay finite dimensional normed linear space $(X,||.||)$ any proper subspace is closed but I am not sure how to…
hmmmm
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