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, as well as measure, ...

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Embedding of Lp spaces

I've managed to prove that for $ 1\leq p < q \leq +\infty $ we have an inclusion (embedding) $ L_q([0,1],\lambda) \rightarrow L_p([0,1], \lambda) ~~ (\lambda $ being Lebesgue measure). The trouble ...
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7 views

Topology of Fréchet space with values in a matrix

Let $F$ be a Fréchet space with seminorm $\{\rho_a\}$ and values in $\mathbb{C}$. I want to consider $\widetilde{F}:=F\otimes \mathbb{C}^{N\times N}$, i.e. a matrix-valued space of $F$, i.e. a ...
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1answer
17 views

Discontinuous mapping between function spaces

Let $ C([0,1]) $ be a space of continuous real-valued functions over interval $[0,1]$ and $ \|f\|_2 = (\int_0^1|f|^2 \, dx)^{1/2} $ define a norm over this space. Prove that the following mapping: $ ...
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12 views

Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
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1answer
12 views

Association of a vector space to metric, normed and inner product spaces

There is a nice visual representation of mathematical spaces from this post: I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but ...
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1answer
20 views

Proving an orthonormal set is an orthonormal basis in Hilbert space [duplicate]

Consider a separable Hilbert space $H$, and $\{g_n\}$ is an orthonormal basis of $H$. Now there is an orthonormal set $\{f_n\}$ that satisfies $\sum_n\|f_n-g_n\|^2<1$. Show that $\{f_n\}$ is also ...
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1answer
9 views

Showing T intertwines $D_T$ and $D_{T^*}$ using Spectral Theorem

Suppose $T$ is a contraction on a Hilbert space $H$ (separable, if you wish). $D_T=(I-T^*T)^{1/2}$ and $D_{T^*}=(I-TT^*)^{1/2}$. I want to show that $TD_T=D_{T^*}T$. I had done this before using a ...
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0answers
16 views

Compositions and products on Sobolev spaces

Does anybody have a good textbook reference for someone who wants to begin studying products and compositions in Sobolev spaces, where the underlying domain is either $\mathbb{R}^n$ or an open subset ...
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11 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists ...
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1answer
43 views

A lemma by Foguel and Weiss [1973]

so I am reading Krengel's text on Ergodic theorems. And the next lemma bugs me as for the proof of it. It's by Foguel and Weiss. Statement: If $P_1, P_2$ are commuting elements of a Banach algebra ...
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30 views

A complicated question

I have the following operator $A: H^1_{0,p}\longrightarrow H^1_{0,p}$ be defined by \begin{equation} Au(t)=\int_0^{+\infty} G(t,s)q(s)f(s,u(s))\,ds-\sum_{k=0}^{+\infty}G(t,t_k)h(t_k)I(u(t_k)), ...
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0answers
36 views

T is not compact operator

I want to show that if $T$ is a bounded operator between two Hilbert spaces and $T$ is not compact then there exists an orthonormal sequence $y_{n}$ and an $R>0$ such that $\forall n\in ...
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0answers
25 views

Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
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1answer
16 views

Help with the proof of the open mapping theorem

I don't understand the following from the proof of the open mapping theorem. Suppose $A $ is a bounded linear transformation from the Banach space $X $ onto the Banach space $Y $. Using Baire's ...
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1answer
43 views

$p$-summable series in a Banach space

Let $E$ be a Banach space and denote its dual space by $E^*$. Let $p \in [1, \infty)$ and $x : \mathbb{N}\rightarrow E$ be such that for every $\phi \in E^*$, $$\left( \sum_{n=1}^{\infty} \lvert ...
2
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1answer
25 views

Are the normed spaces $ \mathbb{R}^{n^2} $ and $ M_n(\mathbb{R}) $ isometric?

Consider the spaces $ \mathbb{R}^{n^2} $ with euclidean norm and $ M_n(\mathbb{R}) $ of $n\times n$ matrices with the norm defined by $ \Vert A\Vert = \sup\limits_{\Vert x\Vert \le 1}\Vert Ax\Vert$. ...
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1answer
22 views

$L^p$ norm of a gradient

Suppose $f:\mathbb{R}^n\to \mathbb{R}$ and let $Df=(f_{x_1},f_{x_2},..., f_{x_n})$, the gradient of $f$. A special case of the Gagliardo-Nirenberg inequality says that $$||f||_{p^*}\leq ...
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1answer
37 views

is it true that $\int\limits_{x_{n}}^xf(z)dz\longrightarrow 0$

If $x_{n}\longrightarrow x$ then is it true that $\int\limits_{x_{n}}^xf(z)\,\mathrm dz\longrightarrow 0$? We have that $f\in L_{2}(0,\infty)$ and takes complex values. I think that it is, but why? In ...
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0answers
20 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
2
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1answer
22 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
2
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1answer
16 views

One Note about One to one and Surjective of linear functional [on hold]

I read a note that: if $ f \neq 0$ is a linear functional on H, then f is onto (surjective) and it is not one to one (injective) in general. Why this is true? i think it need advance ...
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22 views

Equivalent definitions of the trace of a Hilbert-Schmidt operator

I am currently reading the book Spectral Methods in Automorphic Forms, and Iwaniec defines the trace operator in a different way than I am accustomed to. Throughout, assume that everything converges ...
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11 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

Does anyone know a simple proof of the following theorem stating that a positive Borel operator measure $P$ on $\mathbb{R}$ can be written as $V^{\star}EV$ for a Borel spectral measure $E$? ...
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22 views

I need a finite lower bound for this functional, or to prove that one does not exist.

Let $0\leq g < \kappa$, $\gamma>0$ and let $f_1,f_2, S$ be arbitrary functions of r, with $f_1,f_2\geq 0$ I'm looking for a lower bound on the functional $\mathcal{E} = \frac{1}{2} ...
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1answer
19 views

Existence of minimum in $H^{1,2}(\Omega)$

I am considering a functional $$\mu(\Omega) = \min \{ u \in H^{1,2}(\Omega), \frac{\alpha \int_{\partial \Omega} u^2 ds + \int_{\Omega} |\nabla u|^2}{\int_{\Omega} u^2 dx} \}$$ I want to show the ...
5
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0answers
68 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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29 views

Spectral Measures: Spectral Subspaces

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a normal operator $N:\mathcal{D}\to\mathcal{H}$. Denote its associated Borel spectral measure by: ...
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1answer
24 views

If $z$ is in an inner product space $X$, show that $f(x)=\langle x,z \rangle$ defines a bounded linear functional $f$ on $X$.

If $z$ is any fixed element of an inner product space $X$, show that $f(x)=\langle x,z \rangle$ defines a bounded linear functional $f$ on $X$, of norm $||z||$. If the mapping $X\to X'$ given $z\to f$ ...
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40 views

Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
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1answer
55 views

what does it means $∥T(1_{[a,b]})∥$

For each $f\in L_{2}(0,\infty)$, we set $Tf:(0,+∞)\to \mathbb{C}$ with $Tf(s)=\frac{1}{s}\int\limits_{(0,s)}f(t)dt$. For each $0<a<b$ i want to show that $∥T(1_{[a,b]})∥_{2}\geq\frac{b-a}{\sqrt ...
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1answer
31 views

Implications of some sort of $l^2$/uniform convergence

Sorry about the title, but I couldn't really figure out how to describe my problem in one sentence... I'm having some problems with real limits: For $f,g : \mathbb{N} \to \mathbb{R}$ let ...
2
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1answer
24 views

$L^{2}$ convergence, bounded function.

Let $X$ be a metric space and $\mathcal{B}(X)$ be a Borel $\sigma$-algebra on $X$ and $\mu$ be a finite measure on $X$. We consider continuous functions (denoted by $\{f_{n}\}$) on $X$. If $f_{n}\to ...
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1answer
42 views

Analytic skills in applied math

I am an unexperienced undergraduate student just took few basic math classes. And I have found analysis classes really interesting, like basic analysis, measure theory and functional analysis, and ...
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0answers
12 views

$\|\nabla f\|_p\leq C (\|\nabla \times f \|_p +\|\nabla \cdot f\|_p)$

Let $f\colon\mathbb{R^3}\to \mathbb{R^3}$ have compact support. The identity $$ -\Delta = \nabla\times\nabla \times - \nabla \nabla \cdot, $$ and two integration by parts shows that $$ \|\nabla f\|_2 ...
2
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0answers
32 views

T is not compact and orthonormal sequence [duplicate]

I want to show that if $\,T$ is not compact then there exists an orthonormal sequence $x_{n}$ and $R>0$ such that $ \forall n\in \mathbb{N}\,\,\,\,\|T(x_{n})\|\geq R$. It is obvious by the ...
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1answer
43 views

Hilbert Subspaces: ONB

This might be a duplicate. If so, then please let me know. Thanks! Given a Hilbert space $\mathcal{H}$. Consider a dense subspace $\overline{Z}=\mathcal{H}$. Then it provides an ONB: ...
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2answers
47 views

Spectral Measures: Riemann-Lebesgue

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a selfadjoint Hamiltonian $H:\mathcal{D}\to\mathcal{H}$. Denote its associated Borel spectral measure by: ...
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0answers
11 views

Does this gradient map have a closed range?

Let $\mathbb{T}^n$ be $n$-dimensional torus. Let $H^1(\mathbb{T}^n)$ be the Sobolev space of functions in $L^2(\mathbb{T}^n)$ whose weak derivative is in $L^2(\mathbb{T}^n)$. Then the gradient map ...
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16 views
+50

for every frame $B$ with $k$ vector in $\mathbb{R}^2$ such that $B \cup v$ is a tight frame.

Prove the following: for every frame $B$ with $k$ vector in $\mathbb{R}^2$ such that $B \cup v$ is a tight frame. Is the same statement true in $\mathbb{R}^3$? Through the discussion provided in ...
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8 views

A Fredholm alternative for nonlinear operators?

There is a Fredholm alternative of the form: Let $K$ be a compact linear operator. Then $(I + K)u = f$ has a solution $u$ for every $f$ if and only if $$\text{$(I+K)u=0 \implies u=0$.}$$ Is ...
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0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
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1answer
28 views

Show that A is unitary

I'm trying to show that $S+i(I-S^2)^{1/2}$, where $S$ is a self adjoint matrix of norm $\leq 1$, is unitary. I have already checked that $I-S^2$ is positive. I am aware that I need to use the ...
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1answer
17 views

Reducing Subspaces: Nonexample?

Given a Hilbert space $\mathcal{H}$. Consider an operator $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose there exists a closed subspace $Z\leq\mathcal{H}$: $$TZ\subseteq Z,TZ^\perp\subseteq Z^\perp$$ ...
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0answers
38 views

Additivity of Lebesgue integral w.r.t. sets on non-finite domain

I know that for any Lebesgue integrable function $f:X\to\mathbb{C}$, or $f:X\to\mathbb{R}$, where $X$ is a set of finite measure such that $X=\bigcup_n A_n$, $\forall i\ne j\quad A_i\cap ...
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0answers
20 views

Does natrual numbers are isomophic to integers? [on hold]

If there exists a natural numbers algebraic structures , N is the set of natural numbers, which is equipped with the addition operation on it. For another integers algebraic structure, Z is the set of ...
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1answer
7 views

Ordered Projections: Range

Given a Hilbert space $\mathcal{H}$. Consider two orthogonal projections $P,Q$. Then: $$P\leq Q\implies\mathcal{R}(P)\subseteq\mathcal{R}(Q)$$ The ordering being induced by: ...
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0answers
33 views

The operator Tf(x)=1/x∫f(t)dt on L2 is not compact [on hold]

Let $H=L_{2}(0,+\infty)$. For each $f$ define $T_{f}:(0,+\infty) \longrightarrow \mathbb{C}$ with $T_{f}(x)=\frac{1}{x}\int\limits_{0}^{x}f(t)dt$. I want to show that i) $T_{f}$ is continuous ...
3
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1answer
20 views

Sequence of bounded linear functionals on $C^1[0,1]$ that shows Principle of Uniform Boundedness fails without completion.

Let $X$ be the normed vector space $C^1[0,1]$, of continuously differentiable functions on $[0,1]$ with the sup norm $\displaystyle \|f\|=\max_{t\in[0,1]}|f(t)|$. Find a sequence of bounded linear ...
-2
votes
1answer
29 views

a question about contractions on Hilbert spaces

Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$, $\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$. $T_1T^\ast_1+T_2T^\ast_2\leq I$. Then can we verify that 1 ...
1
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1answer
30 views

In a Hilbert Space, if $\langle x,x_n \rangle \to 0$ then $\sup\{\|x_n\|:n=1,2,3,…\}<\infty$

Let $\mathbb{H}$ be a Hilbert space. Let $\{x_n\}$ be a sequence in $\mathbb{H}$ with the property that $\langle x,x_n \rangle\to 0$ as $n\to\infty$ for $x\in\mathbb{H}$. Show that ...