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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).

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3 views

n-times commutator of a function with $-\Delta +v$

Let $f$, $v$ be smooth function and let us assume that $v$ is also bounded. I'm needing somehow a formula for the $n$-times commutator of $f$ (interpreted as multiplication operator) with the ...
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0answers
6 views

Spectral decomposition of the resolvent map

Let $P_\Omega$ be a projection valued measure and let $R_A(z)=(A-z)^{-1}$ be the resolvent map. It can be shown that $$R_A(z)=-\sum_{j=0}{\frac{A^j}{z^{j+1}}}$$ whenever this series is defined. My ...
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0answers
10 views

Michael's selection theorem

Michael's selection theorem states that a lower hemicontinuous multivalued map with nonempty convex closed values $\displaystyle F\colon X\rightrightarrows E$ from a paracompact space $X$ to a Banach ...
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1answer
26 views

$X,Y$ Banach, $V \subset X$ a linear subspace, $T:V\to Y$ closed, then $T$ bounded $\iff$ $V$ closed

Let $X$ and $Y$ be Banach spaces and let $V \subset X$ be a linear subspace. Let $T: V \to Y$ be a closed linear operator. Show that $T$ is bounded if and only if $V$ is closed. For the direction $V$ ...
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1answer
24 views

Proof no nonnegative function satisfies three norm-like integrals

$V$ is the real inner product space $C_\mathbb{R}[0,1]$ and $a \in [0,1]$. Prove there is no nonnegative function $f \in V$ such that $$ \int_0^1 f(x) \, dx = 1, \int_0^1 xf(x) \, dx = a, \mathrm{and} ...
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1answer
45 views

Question about a “w.l.o.g.” statement in proof of Uncertainty Principle?

I read a proof on the Uncertainty Principle (see below) and although the technical part itself is relatively straight forward I still do not understand a certain "w.l.o.g." statement in the proof. ...
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1answer
23 views

Is tempered distribution a distribution?

In Distribution Theoryand Fundamental Solutions of Differential Operators we can find following theorem with proof. I think this proof is not valid starting from blue sentence. To show that $f\...
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2answers
32 views

If $U\subset W\subset V$ and $W\cap U^\perp = \{0\}$ then $U=W$

Statement : Let $V$ be a Hilbert space. Let $U\subset W\subset V$ be closed subspaces. Suppose that $W\cap U^\perp = \{0\}$. Then $U=W$. I know this is true in the finite dimensional case (see proof ...
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0answers
11 views

are these finite-dimensional spaces uniformly complemented in lp?

Let $(x_i)_{i=1}^\infty$ be a Schauder basis for a Banach space $X$, and let $1\leq p<\infty$. We say that $\ell_p$ contains uniform copies of $[x_i]_{i=1}^N$ if there exists a constant $C\in[1,\...
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0answers
10 views

Constant of continuity as a monotonically increasing function of the radius of the neighbourhood

While reading the fairly famous paper by Brezzi et al from 1980 (link), I got slightly puzzled by one of the conditions, in particular in Theorem 1 (Implicit Function Theorem basically). They consider ...
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1answer
47 views

On the Schrodinger fundamental solution

Let $e^{it\Delta}$ be the fundamental Schrodinger solution. If $u_0$ is the corresponding initial data to the problem associated to Schrodinger free equation $u_t = i\Delta u$ and $S(\mathbb{R}^N)$ ...
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1answer
26 views

If $T(t)$ is an immediately differentiable semigroup on $H$ with generator $A$, does $\frac{d}{dt}\|T(t)x\|_H^2=2⟨AT(t)x,T(t)x⟩_H$ hold for all $x∈H$?

Let $(T(t))_{t\ge0}$ be a semigroup on a $\mathbb R$-Hilbert space $H$ with $$\sup_{s\in[0,\:t)}\left\|T(s)\right\|_{\mathfrak L(H)}<\infty\tag1$$ for some (and hence all) $t>0$ and $(\mathcal D(...
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1answer
21 views

Maybe “misleading” use of $\liminf$ in proof of $L^{\infty}$ completeness (from Linear Functional Analysis by Hans Wilhelm Alt, p.52)

I have come across suspicious use of liminf a few times, while reading the book. So I decided to post one short case because it seems to me I am probably understanding it wrong. For me limsup does ...
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0answers
13 views

Calculus of Variations by Charles Fox: Question on Statement in Section 2.4

Fox states in Section 2.4, pg. 38, that "Anticipating this result, it follows that even if $u(x)$ vanishes at either or both of the values $x=a$ and $x=b$, both $t^2(a)/u(a)$ and $t^2(b)/u(b)$ still ...
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2answers
14 views

Geometric proof for Composition bound property of operator norms?

This is just a curiosity. For linear transformations $A$ and $B$, $||AB|| \le ||A|| \cdot ||B||$ where$||\cdot||$ denotes the operator norm (Of course provided $AB$ exists.) This fact has a proof, but ...
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1answer
14 views

GNS representation of a nuclear $C^*$-algebra

Suppose $A$ is a nuclear $C^*$-algebra with a tracial state $\psi$, $(\pi_{\psi},H_{\psi})$ is the GNS reprsentation with respect to $\psi$. My question: Does there exist $A$ which satisfy the above ...
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1answer
25 views

Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}

Here we use infinity norm as metrics for $C[0,1]$. The professor claims that this set is compact. I can show this set is relative compact by Arzela Ascoli, i.e., for each subsequence there is a ...
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0answers
24 views

Are there degrees or categories of orthogonality?

The vectors $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$ are orthogonal under the dot product. The vectors $1$, $...
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1answer
10 views

Are $L_p$ norm and discrete $L_p$ norm comparable?

Are there any estimates on how a $L_p$ norm (say for a compact set in $\mathbb{R}$) is related to a discrete $L_p$ norm, where we could for example consider the Jackson integral on this compact set. ...
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0answers
25 views

What is the minimal requirement for a Fourier transformation?

I have a lot of comprehension questions that I can't really figure out by googling: The difference between a Fourier transformation and a Fourier series. Am I correct in thinking that the difference:...
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0answers
25 views

Adjoint of Bounded Below Operator

Let $H$ be a complex, separable Hilbert space, and $T:H \rightarrow H$ a linear, bounded operator. Assume that $$\sigma(T) = \sigma(T^*) = \{ \lambda \in \mathbb{C}: a \leq |\lambda| \leq b \}$$ for $...
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1answer
33 views

How to define $L^{\infty}$ norm for vector-valued functions?

for functions $f: \mathbb{R}^{n} \rightarrow \mathbb{R} $ and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$, $\| f-g\|_{\infty} := \sup_{x\in \mathbb{R}^{n}}\|f-g \|$. But if now we have $f: \mathbb{R}^{...
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0answers
18 views

Different topologies in Sobolev space $W^{1,p}$

In paper [1], L.Ambrosio talks about the space $W^{1,p}(\Omega),\ 1\leq p<+\infty$ endowed with four different topologies: The strong topology, denoted by $W^{1,p}(\Omega)$. The weak topology, ...
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0answers
22 views

Proving one form of Ito Isometry using Functional Analysis

I would like to know whether it is possible to give a proof of (one form of) Ito Isometry using a tool which I like to call "the functional analysis"-way. Let me explain the settings first. What we ...
2
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0answers
39 views

Two PDE for one matrix-valued unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $P = P(x,t) \in \mathbb{R}^{3\times 3}$, $Q = Q(x,t) \in \mathbb{R}^{3\times 3}$, $R_0 = R_0(x) \in \mathbb{R}^{3\times 3}$ and $G= G(t) \in \mathbb{R}^{3 \...
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0answers
23 views

Extension of Choi's theorem

In the paper written by Man Duen Choi "Completely Positive Linear Maps on Complex Matrices", there was a criteria mentioned/theorem. For reference I have written it below. Let $ϕ:M_n→M_m$. Then ϕ is ...
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0answers
14 views

Weak star convergence of Borel probability measures on a metric space

Let $(X,\rho)$ be a compact metric space and let $P(X)$ be the set of Borel probability measures on the Borel $\sigma$-algebra of $X$. Suppose $\mu_n,\mu \in P(X)$ for $n \in \mathbb{N}$ such that $\...
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0answers
29 views

The span of a finite number of vectors in a normed vector space is closed

I want to prove that the span $S$ of a finite number of vectors $v$ in an arbitrary normed vector space $V$ is a closed set. My plan is to show that all convergent sequences ${x_n}$ contained in the ...
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0answers
16 views

Simple Consequences of Goldstine's Theorem

For a normed space $ X $, let $ J : X \to X^{**} $ be the natural embedding of $ X $ into $ X^{**} $, and let $ B_X $ and $ B_{X^{**}} $ denote the closed unit balls of $ X $ and $ X^{**} $ ...
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1answer
22 views

Absolutely continuous Banach space valued function

Let $X$ be a Banach space and $F:[a,b] \to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f \in L^1([a,b],X)$, is the ...
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0answers
26 views

Landau inequality for several variables

For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$ \|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ ...
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1answer
32 views

If operator is closed and densely defined then $D(A^*)^\perp = \{0\}$

I'm a bit rusty in my Functional Analysis and couldn't solve this question: Let $X$ be a Banach space (over either $\mathbb{R}$ or $\mathbb{C}$) and $X^*$ its dual space. Show that, if $A:D(A) \...
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0answers
30 views

Why is $ \max_{i} | \lambda_i(A) | \leq \| A \|_P $?

I was told: $$ \max_{i} | \lambda(A) | \leq \| A \|_P $$ I tried thinking through it. So the operator norm is defined as: $$ \| A \|_P = \sup_{y \neq 0} \frac{ \| A y \|_P }{ \| y\|_P } = \sup_{ \| ...
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0answers
26 views

Rewriting a $\max\left\{0,\dots\right\}$ function in order to integrate the function more properly

Yesterday I asked a question about a certain integral. In the integral is the term: $$\max\left\{0,\left|\text{n}\cdot\sin\left(2\pi\cdot x\cdot t-\frac{\pi}{2}\right)\right|-2\cdot\text{z}\right\}\...
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0answers
40 views

Couldn't Derive the Result Given in a Book

Currently I'm trying to read Stroock's "lectures on topics in stochastic differential equations" book. In lemma $2.5$, he states that given $f$ and $g$ that are strictly increasing function on $[0, \...
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2answers
55 views

Uniqueness in $C([0,T])$ of solution found by Picard-Lindelof Theorem

Full statement of the question: Suppose $\alpha \in \mathbb{R}$, also $f: \mathbb{R} \rightarrow \mathbb{R}, f \in C^{1}(\mathbb{R}), and f(0) = 0$ Consider the following ODE: $\partial_{t} u(t) =...
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0answers
22 views

Difficulty Understanding Sufficient Conditions for Weak Extrema in Calculus of Variations

I am having a difficult time understanding Jacobi's necessary condition for weak extrema of functionals. Graphics and detailed explanations would be helpful. I am following the following two texts: ...
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1answer
37 views

If $ f*g$ is a polynomial of degree at most $m$ for all $g \in C_{0}^{\infty} $. Show $f$ is a polynomial of degree at most $m$ almost everywhere

$C_{0}^{\infty}$ denotes the set of smooth functions with compact support. In attempt of this, I've evaluated the convolution at the (m+1)th derivative to obtain $$ \frac{d}{d^{m+1}}(f*g) =\int f(x-y)...
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0answers
22 views

Weak convergence and compacity

Please I dont understand this. I have: $ \parallel \nabla m_n \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C$ $ \parallel \frac{\partial m_n}{\partial z} \parallel_{L^{\infty}(\mathbb{R}^+,...
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1answer
41 views

Uniform inequality for a continuous function

Let $f(x,y)\in \mathcal{C}([a,b]\times[c,d])$ such that $$\exists \xi\in (a,b) : f(\xi,y)\neq 0, \forall y\in [c,d].$$ By the continuity of $f$, we have $$|f(\xi,\cdot)|\geq \min\limits_{[c,d]} |f(\...
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0answers
32 views

What is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
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0answers
32 views

For a compact Riemannian manifold $M$, $L^2(M)$ is spanned by the eigenfunctions of the Laplacian.

In some paper I read the following statement: For a compact Riemannian manifold $M$ and the corresponding Laplace-Beltrami operator $\Delta$ on $M$ we have, that $$L^2(M) = \widehat{\bigoplus_{\...
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0answers
18 views

Shift operator on the double-size Hilbert space $\ell^2(\mathbb{N}^*)\oplus \ell^2(\mathbb{N}^*)$

It is well known that, the right shift operator is given by \begin{align*} A_1\colon \ell^2(\mathbb{N}^*) & \rightarrow \ell^2(\mathbb{N}^*) \\ (x_1,x_2,\cdots)&\mapsto (0,x_1,x_2,\cdots), \...
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1answer
15 views

Show that resolvant is analytic outside the spectrum

Let $T$ be a bounded operator on $Hilbert$ space $\mathcal{H}$. Show that $R_{\lambda}=(T-\lambda)^{-1}$ is an analytic function on open set $\rho(T)=\mathbb{C}\setminus sp(T)$. I know $R_{\lambda}=-\...
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1answer
35 views

The tensor product of two blocks of positive operators is positive

Let $$T = \begin{bmatrix} T_{11} & T_{12}\\ T_{21} & T_{22} \end{bmatrix},\quad S = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix}$$ be two positive operators on $E\...
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1answer
50 views

Prove $\|T\| = \sup_{\|x\| < 1} \|Tx\|$

Let $X, Y$ be Banach spaces. And $T \in B(X\rightarrow Y)$. Prove that $$\|T\| = \sup_{\|x\| < 1} \|Tx\|$$ Discussion Having trouble seeing how to handle some of these ideas below. Please let me ...
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0answers
19 views

Weak Convergence from Strong Convergence

Let $\Omega \subset \mathbb{R}$ be a bounded domain, $v \in H_{0}^{1}(\Omega)$ such that $||u_{n}-v||_{H_{0}^{1}(\Omega)}\to 0$ as $n\to\infty$ for a bounded sequence $\{u_{n}\}_{n\in\mathbb{N}} \...
3
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2answers
40 views

Deforming a linear map a little preserves surjectivity

Let $X$ be a Banach space, and $A: X\to X$ be a surjective linear map. Define $$\eta(A) = \{\lambda\in \mathbb{C}: A - \lambda I \text{ is surjective}\}$$ where $I: X\to X$ is the identity. Show that $...
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2answers
48 views

What is the Geometric meaning of vector norm in Rn n>3

My question is related to the length of the vector , Sorry it may seem stupid for you as i come from engineering background not mathematics background For Vectors up to 3 dimensions (can be ...
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1answer
34 views

An identity for Fourier transform of measure

Consider a finite Borel measure $\mu$ on $\mathbb R$. The Fourier transform $\hat{\mu}$ of $\mu$ is defined by $\hat{\mu} (\xi)= \int _{\mathbb R} e^{-ix\xi} d\mu(x)$. I would like to prove the ...