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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 ...

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Uniformly approximating $f\in \mathcal{C}([1,\infty))$ with polynomials where $\underset{x\rightarrow +\infty}{\lim} f(x)=a$

Suppose $f\in \mathcal{C}([1,\infty))$ and $\underset{x\rightarrow +\infty}{\lim} f(x)=a$. Show that $f$ can be uniformly approximated on $[1,\infty)$ by functions of the form $g(x)=p(1/x)$, where $p$ ...
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Remark 16 in Chapter 2 from Brezis - Functional Analysis, Sobolev Spaces, and Partial Differential Equations.

Let $E,F$ be two Banach Spaces and $A : D(A) \subset E \to F$ be a linear unbounded operator which is densely defined. Now, we would like to define $A^{*}$ as the adjoint of $A$. Let $A^{*} : D(A^{*}) ...
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Value of operator norm when $\mathcal{T}f(x)=\int^{x}_{0} f(t)dt$

Let $\mathcal{C}$ be the space of continuous functions on $[0,1]$ equipped with the norm $\|f\|=\int^{1}_{0}|f(t)|dt$. Define a linear map $\mathcal{T}:\mathcal{C}\rightarrow \mathcal{C}$ by $$ \...
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Iteration of a sequence

I am reading a paper on Piecewise Convex Transformations. I got an inequality as : $||P_{\tau} f||_{\infty} \leq {\frac{1}{\alpha}|| f||_{\infty} + C||f||_{1}}$ After n iterations, we obtain : $||P_{...
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Disambiguation and Clarification of Adjoint Operators

I want talk about something that has been bugging me today regarding adjoint operators on finite dimensional inner product spaces. So here has been my understanding, let $T$ be an operator on $V$; $V$...
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Does we have separation theorem for closed subsets in a topological vector space?

In Rudin's Functional Analysis, page $10,$ he stated the following separation theorem for topological vector space. Theorem $1.10:$ Suppose $K$ and $C$ are subsets of a topological vector space $X,$...
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Balls in finite dimensional normed spaces

Let $X$, $Y$ be normed linear spaces and $Y$ be finite dimensional. Suppose $T \in B(X,Y)$ such that $T$ is surjective. Prove that there is some $\delta > 0$ so that $B_\delta(0_Y ) \subseteq T(B_1(...
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32 views

Factorization for $e^{\lambda x}$

Let $\lambda, x$ be real numbers. Why can't we factorize $$e^{\lambda x}=f(\lambda)g(x)$$ for some functions $f$ and $g$?
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16 views

Neighborhood of zero on weak topology

I have a homework, I have tried so much on this question, but I couldn't solve it. How can I prove it? Can you help me? Thank you. $E$ is a normed space and $E'$ is the topological dual of $E$. If $...
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existence of isonormal Gaussian process with non-separable Hilbert spaces

Let $H$ be a non-separable real Hilbert space. Does there exist an associated isonormal Gaussian process $W$ ? I know that the answer is yes for separable real Hilbert space.
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Hypercyclic operators in $L_p (0,\infty)$

I'm pretty new to this area of study so if there are logical lacune in my proof (I'm sure there are many) please let me know. This is material I'm self studying. I'm trying to adapt the methods used ...
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Finding operator norm of functional

I am given a functional $F:C([0,1],\|x\|_\infty) \to \mathbb C $ by formula $$F(f)=2\int_0^{1/2}f(t)dt \text{ - } \int_{1/2}^1f(t)dt$$ I should find its operator norm. I've found that $\|F\| \le 3/2$...
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Infinite dimension - convergent [on hold]

In finite dimensional normed space componentwise convergence implies convergence. Converse is true. But in infinite dimensional normed space, componentwise convergence does not implies convergence. ...
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16 views

Property of supremum for any real exponent

For any $\alpha>0$ and $f:\Omega\to(0,\infty)$ bounded, is it true that $\sup f^{\alpha}\leq (\sup f)^\alpha$? Also what about $\alpha<0$. I know for $\alpha\in\mathbb{N}$, it holds. But for ...
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Equivalence of maximum in thenunit circle and null function on the basis

Given $\{e_n\}$ the unit basis of $c_0$ show that for $f \in c_0 ^*$ its equivalent: The function $|f|$ have, $m$ in the unit circle of $c_0$ Existe $m$ in $\mathbb{N}$ such that $f(e_n) = 0$ for all ...
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Steps of Verify Completeness of a Normed Vector Space

The textbook by Kreyszig on functional analysis identifies three steps to verify the completeness of a space: Construct an element to use as a limit: $x$ Prove that $x$ is in the space considered ...
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33 views

Inequality in $L^2$

Let $u,v\in L^2(\mathbb{R}^d)$, I want to prove the following ineqality $$\|u\|^2_{L^2(\mathbb{R}^d)}\ge a \|v\|^2_{L^2(\mathbb{R}^d)}-\|u-v\|^2_{L^2(\mathbb{R}^d)}$$ for all $u,v\in L^2(\mathbb{R}^...
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Question on Sobolev extension to boundary

Let $U \subset \mathbb R^3$ be an open, bounded and connected set with a $C^2-$regular boundary $\partial U$. I'm trying to understand the following implication: If $f\in W^{1-1/2,2}(U)$ then $f{\...
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17 views

Continuity of sublinear functional on cone

Let $E$ be a normed space. If $p\colon E\longrightarrow \mathbb{R}$ is sublinear and continuous at $0$, then $p$ is continuous. The proof is pretty straightforward, one just has to observe that ...
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Linear operator in Lp and its norm

Given $$g \in L_\infty[0,1]$$ show that, for 1 ≤ p ≤ ∞, $$ f \rightarrow f\cdot g$$ is a linear continuous operator of $$L_p[0,1] \to L_p [0,1]$$ and compute its norm
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Is this differential operator Hermitian?

The operator is $$\hat{A} = -i \left(x \frac{d}{dx} + \frac{1}{2} \right).$$ Is it true that $$\langle \hat{A} \psi_1(x)|\psi_2(x)\rangle = \langle \psi_1(x)|\hat{A}\psi_2(x)\rangle\ ?$$ Here, $\...
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pairwise orthogonal projections in an inseparable $C^* $ algebra

If $A$ is a separable $C^*$ algebra,then there are at most countable pairwise orthogonal projections.If $A$ is inseparable,how many pairwise orthogonal projections in $A$? If it has, is it ...
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31 views

Estimate on average with weight

Let $B_{2R}=B(x,2R)$ be the ball of radius $2R$ centered at $x$ and $v=log\,u$ for some positive function $u$ defined on $B_{2R}$. Denote by $v_{B_{2R}}=\frac{1}{w(B_{2R})}\int_{B_{2R}}v(x)w(x)\,dx$, ...
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Spectral density of stochastic partial differential equations

I am trying to understand some examples of computing the spectral density of SPDEs. The root of my confusion is that some terms appear to be missing from the calculation of the squared transform, and ...
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1answer
23 views

Proving uniform boundedness

Let $-\infty < a<b<\infty$ and $A \subseteq C([a,b]) \cap C^1((a,b))$ satisfying $\forall f\in A$ $\int_{b}^{a} |f(x)| dx + \int_{b}^{a} |f'(x)|^p dx < C$, where $C>0$ and $p>1$. I ...
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19 views

Proof Understanding: E Normed space, F Banach space implies $L(E,F)$ is a Banach Space..

I am working through Hueser's Functional Analysis and have come across the proof (7.4) that: If E is a normed space and F is a Banach space then $L(E,F)$, the set of continuous linear transformations ...
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39 views

Working with this 'almost everywhere' statement

Consider a real function $f$ and take $\mathbb R$ together with Lebesgue measure $m$. I want to check if I have the right reasoning with regards to working with the following "almost everywhere" ...
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Integral in termsof supremum

I am facing difficulty to prove the following fact: If $w$ is a locally integrable positive function in $\Omega$, then $$ \sup_{B(x,R)}\lvert v\rvert=\lim_{p\to\infty}\lVert v\rVert_{L^p(B(x,R),w))}, ...
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Dual spaces of $\ell^2$ of an arbitrary normed space

Let $(V,\|\cdot\|_V$ be a normed space, and consider $\ell^2(V)$ to be the standard $l^2$ space using sequences in $V$ (instead of $\mathbb{R})$. We can consider the dual space pair $(V,V')$. We know ...
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Problem 4 Barry Simon a comprehensive course in analysis part 1.

(a) For any bounded Baire function, $f$, on a compact Hausdorff space, X, prove that $||f||_{\infty}:=\inf\left\{\sup_x |g(x)|: f-g=0\text{ for a.e. } x\right\}$ exists, defines a seminorm, and equals ...
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Norm on semi vector spaces possible or trivial?

Because i always see positives spaces over positive semi fields and nowhere it has anything negative?So is it a trival case or not necsessary to define norm ?Since norm makes everything positive
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Compact operators on $\ell^1$

Let $T\in \ell^1$, $Tx = (\lambda_1x_1,\dots,\lambda_nx_n,\dots)$. Want to show that if $T$ is compact, then $\lambda_n\to0$. I know for $p\in(1,\infty]$, canonical basis $e_n \rightharpoonup 0$ (so ...
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Property of continuous functions defined on a sphere of R^n where n is odd

If $B=B(0,1)$ is the unit open ball in $\mathbb{R}^n$ with $n$ odd and $f:\partial B \rightarrow \partial B$ is continuous, then exists $x\in \partial B $ such that $f(x)=x$ or $f(x)=-x$. Any idea of ...
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spectral properties of the shift operator

Let $E = \ell^2$, and consider the multiplication operator $M$: $$Mx = (\alpha_1x_1,\dots,\alpha_nx_n,\dots), \forall x\in E$$ where $(\alpha_n)$ is a bounded sequence in $\mathbb{R}$. And consider ...
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If $f_k \to f$ uniformly, then $J(f_k) \to J(f)$, where $J: \mathcal{C}_{\text{c}}(\mathbb{R}^d) \to \mathbb{R}$ ist linear and monotonous

Let $\mathcal{C}_{\text{c}}(\mathbb{R}^d) := \{f: \mathbb{R}^d \xrightarrow{\mathcal{C}} \mathbb{R}: \text{supp}(f) \text{ is compact.} \}$. I have a few questions regarding the proof following ...
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Stieltjes integral for monotonically increasing functions.

Let $$Z:[0,\infty)\rightarrow[0,\infty)$$ $$K:[0,\infty)\rightarrow[1,\infty)$$ monotonically increasing cadlag functions with $K(0)=1$ and $Z(\infty)<\infty$. I want to show that following ...
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1answer
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Equality of measures on $C([0,1])$, if integrals over functions continuous w.r.t. pointwise convergence coincide

The problem I pose below is a simplified version of my actual problem: Instead of $C([0,1], \mathbb{R})$ I actually have $C([0,+\infty], H)$, where $H$ is some separable, (infinite-dimensional), real ...
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1answer
18 views

$T$ has not a closed range

Let $$T = {\rm diag}(0, 1, 0, \frac{1}{2!}, 0, \frac{1}{3!}, \dots)$$ Clearly $T$ is a positive operator on the Hilbert space $\ell_{\mathbb{N}^*}^2(\mathbb{C})$. I want to prove that $T$ has not ...
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Prove these two functions are in $H^2(\mathbb{R}^n)$

I want to prove that the equation $$u -\Delta u=f$$ with $f \in L^2 (\mathbb{R}^n)$ admits a solution in $H^2 (\mathbb{R}^n)$ with $n=1,3$. Taking Fourier transforms and resolving, I get: $$u(x)=\...
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Energy equalities and estimates for weak solutions

Basically my question is why, when we go from classical solutions to weak solutions, we have to use energy inequalities instead of equalities. More preciscely, consider the Navier-Stokes equations \...
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To verify Lipschitz continuity of the given function $f$.

Consider the function $$f(x,\vec{v}):=g(I_t+\nabla\cdot(I\vec{v}))$$ where $I=I(\vec{x},t)$ is the image intensity function, $g:\mathbb{R}\to[0,\infty)$ is continuous and non negative, $\vec{v}\in H^1(...
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40 views

Closure of the orthogonal complement.

Let H be a Hilbert space and let $A \subset H$. Let the orthogonal complement of A be: $A^\perp$ = {$x \in H : x \perp A$}. How do I show that $A^\perp$ is a vector space and that it is closed? I ...
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Laplace transform of exponential distributed random varibale

Let $T\sim Exp(\lambda)$. I want to calculate the laplace transform $$E[e^{-\delta T}]$$ So far I get $$E[e^{-\delta T}]=\int_0^\infty e^{-\delta x}\lambda e^{-\lambda x}dx=\lambda\int_0^\infty e^{-(\...
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1answer
52 views

How to show non existence of an operator $A: V \to V^*$

Let $V$ be a Banach space , $p \in (1,2)$ and $\mu >0.$ How can I prove that there does not exist an operator $A:V \to V^*$ such that $$\langle Au-Av, u-v\rangle \geq \mu \Vert u-v \Vert ^p,\qquad ...
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1answer
22 views

Operator Norm Question

Suppose I am interested in operators $T:X\to Y$, with $X$ and $Y$ both separable Hilbert spaces. The operator norm of such $T$ can then be taken as $$ \|T\| = \sup_{\|x\|_X\leq 1}\|Tx\|_Y. $$ Since ...
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1answer
41 views

Question regarding motivation of spectral theorem for unitary operators

$\newcommand{\mc}{\mathcal}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ Theorem B.4 in Einsiedler and Ward's [EW] Ergodic Theory with a view towards Number Theory states the following: Theorem 1. ...
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30 views

A sufficient condition for subdifferentiability

Corollary 9 in here (page 31) states that a proper convex function $g:Y\rightarrow \mathbb{R}\cup\{\infty\}$ (not necessarily continuous) on a locally convex space $Y$ is subdifferentiable on a point $...
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1answer
22 views

Does a continuous function embed a separable space into a separable closed subspace?

Suppose that $X$ is a topological space and $Y$ is a normed vector space, and $f:X\rightarrow Y$ is continuous. In general we know that if $X$ is separable, then the image $f[X]$ will be separable ...
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1answer
11 views

Linear operator T on space with inner product on C

I have linear operator $T\rightarrow X:X$ , X is space with inner product on $\mathbb{C}.$ Is my train of thoughts correct? $2 (\langle Tx,iy \rangle + \langle Tiy,x \rangle) = 2(-i \langle Tx,y \...
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1answer
21 views

Reducing subspaces of unilateral shift on $l^2$

Show that there is no non-trivial reducing subspace of the unilateral shift $T:l^2\to l^2$ given by $T((x_1,x_2,...))=(0,x_1,x_2,...)$. So suppose there is a non-trivial reducing subspace $M$ then we ...