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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Initial topology

I have read that for all $U \in \sigma (X, {f_i, i \in I})$ (initial topology) there exists a finite number of open sets $V_i$ in $T_{Y_i}$ s.t. $U = \cap_{i\in G}f_i^{-1}(V_i)$ or $U = \cup_{i\in ...
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13 views

Numerical range of closure of operator

Let $B$ be an unbounded densely defined and closable operator. If $\mathcal{N}(B)$ is the numerical range, what can be said about the numerical range of its closure $\overline{B}$?
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33 views

The dual of $w^{\star}$ topological space

There is a result in my book states: Suppose that $(X, \tau)$ is a locally convex topological vector spaces. $(X, \tau)^{\star}$ is the dual of $(X, \tau)$ and denoted by $X^{\star}$. Let ...
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1answer
51 views

Decoding / reverse engineering math content

I came across the following image while browsing internet: Is this some kind of coded message? Or it has some real math meaning? Or it is just a joke? My math knowledge is way below the level ...
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16 views

Minkowski functional application [closed]

How minkowski functional are applicable in the study of contact angle over irregular surfaces. Please help me by sharing some references if you have.
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13 views

Causal mappings over $C[0,1]$ [closed]

Let $X=Y=C[0,1]$. A mapping $T:X\to Y$ is called causal if for every $r\in[0,1]$ and every $f,g\in X$ if $f(t)=g(t)$ for every $t\in [0,r]$. Then $Tf(t)=Tg(t)$ for every $t\in[0,r]$. 1) Prove that ...
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16 views

Question on the space of all square summable functions involving operator norms and eigenvalues.

Recall that for a set M, $ \mathscr ℓ^2 (M) $ is the space of all square summable functions M $ \to \Bbb C $ . Let $\mathrm T \in Hom( ℓ^2 (\Bbb N) , ℓ^2 (\Bbb N)) $ be given by $$ (\mathrm T a)(n) ...
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21 views

Weak convergence in $l_1$ if and only iff $\|x_n-x\|\to 0$ [closed]

In $l_1$, $x_n$ converges weakly to $x$ if, for every $y\in l_\infty$, $lim_{n\to\infty}\langle x_n,y\rangle=\langle x,y\rangle$ where $\langle x,y\rangle = \sum^\infty_{n=1}x_ny_n$. Prove that in ...
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1answer
25 views

Continuity with restrictions

Suppose that $f \colon A \to \mathbb{R}$ is a function and that $B \subseteq A$. We define the restriction of $f$ to $B$ to be the function $f|_B B \to \mathbb{R}$ defined by $f_B(x) = f(x)$ for all ...
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1answer
42 views

The question regrading to density argument in analysis.

I know the density argument in $L^p$ space, in Sobolev spaces, and even in BV are really sweet in many many cases. However, some times author just work on nice functions and comment by "the rest can ...
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3answers
132 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
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17 views

Compact operator on non complete space - spectrum

I was trying to google it out but without succes. Is the spectrum of compact operator $f:X \rightarrow X$ at most countable in general only if $X$ is banach space ?
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15 views

Existence and uniqueness of SDEs depending on the expected value?

I was thinking of general mean-field SDEs. But let us just look at something really simple: $$dX_t = dt + dB_t, \quad X_0=x$$ the solution to this SDE exists in a strong sense and is: $X_t = x + t ...
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1answer
32 views

Multiplication operator on $L^2$ and spectral theorem.

Let's consider the multiplication operator by the independent variable in $L^2(\mu)$, where $\mu$ is a borel regular measure on $\mathbb{C}$: $Mf(z)=zf(z)$. I want to show that if $\phi$ is a borel ...
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1answer
59 views

A base of topology

Consider a space of smooth functions $C^{\infty}[a,b]$ and a set $$\tau=\left\{B(f,\varepsilon_0,\varepsilon_1...\varepsilon_r):f\in C^{\infty}[a,b],r\in\mathbb{N}\right\} $$ where $f$ is arbitrary ...
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2answers
34 views

Spectral Measures: Support vs. Norm

Given a complex Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: $$T:=\int_\mathbb{C}zdE(z)$$ ...
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3answers
39 views

Spectral Measures: Support vs. Spectrum

Given a complex Hilbert space $\mathcal{H}$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: ...
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0answers
6 views

Separability of Frechet space of m-times continuously differentiable functions

For each $m\ge 1$ and compact subset $Q$ of $\mathbb{R}^{d_1}$, let $C^m(Q,\mathbb{R}^{d_2})$ be the separable Banach space of all $m$-times continuously differentiable functions ...
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1answer
27 views

Spectrum of multiplication operator by the independent variable in $L^2$

If $\mu$ is a regular Borel measure on $\mathbb{C}$ with compact support $K$, define $N_\mu$ on $L^2(\mu)$ by $N_\mu f=zf$ (the multiplication by the indipendent variable). An exercise in "Conway" ...
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1answer
44 views

An exercise (about positive elements) in C*-algebra

Let $A$ be a C*-algebra, $a\in A$ be a positive element and $b\in A$ be an arbitary element in $A$. Can we verify that $$b^{*}ab\leq \|b\|^{2}a~~?$$
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50 views
+100

Solution of parabolic PDE system

For the following parabolic PDE system, $u(x,t)$ and $v(x,t)$ are functions of independent variables $x$ and $t$, $x\in[a,b]$. \begin{equation} \begin{cases} \frac{\partial}{\partial ...
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3answers
38 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
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2answers
20 views

Closure of linear subspace in Topological vector space

Let $X$ be a TVS, $x\in X$ and $M<X$ be a linear subspace. Does $x\in M+U$ for every open neighborhood $U$ of $0$ imply that $x$ is in the closure of $M$? EDIT: This argument is used here: ...
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1answer
31 views

Topological Vector Space not induced by Metric

Can anyone give me an example of a Topological Vector Space that is not metrizable? I know that the neighborhood base of $0$ needs to be incountable, and all I can construct then is no topological ...
3
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1answer
59 views

Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$.

Let $\mathcal{H}$ be a complex Hilbert space, and let $A$ be a bounded operator linear operator on $\mathcal{H}$ with $\|A\| \le 1$. It is known that $\|p(A)\|\le \sup_{|z|=1}|p(z)|$ for all complex ...
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0answers
16 views

Nondegenerate representation

By the definition, we say a representation $(\pi,H)$ is nondegenerate if $cl[\pi(A)H ]= H$. Below I have two theorem, the first from Conway's Functional analysis and the second from Takesaki's ...
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1answer
26 views

Spectral measure and commutativity.

I want to prove that if $A\in B(H)$ and $N\in B(H)$ is a normal operator, and $AE(\Delta)=E(\Delta)A$, where $E$ is the spectral measure given by $N$ and $\Delta$ is a Borel subset of $\sigma(N)$, ...
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21 views

Bounding and continuity in Banach spaces [closed]

Prove that if mapping in Banach space is bounded, it is continuous.
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21 views

Proving that Riesz map is bijection [closed]

1) Prove that Riesz map is bijection 2) Prove that Riesz map is monomorphism
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31 views

Is it possible to find the norm fuction of a space from an inner product already defined for it?

I'm a noob on the subject of functional analysis. As the title of the question says: Is it possible to find the norm fuction of a space from an inner product already defined for it? e.gr.: Suppose ...
2
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1answer
39 views

How to show $l^\infty$ is larger than $ c_c$?

Let $l^\infty$ be the space of bounded sequences and $c_c$ be the space of sequences that only finitely many terms are non-zero. How can I show that there does not exist a bijective linear ...
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55 views

Complex Measures: Lebesgue Decomposition

Disclaimer: This thread is related to: Singular Continuous Measures: Definition? Context Let $\Omega$ be a measure space with finite measure $\mu<\infty$. Consider a finite measure ...
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32 views

Fix point theorem for measures? metric on space of measures?

I have the following problem: I consider a probability space $(\Omega, \mathcal{F}, \mu)$ where $\Omega= C_0([0,1])$ (space of continuous functions on $[0,1]$ starting from 0), $\mathcal{F}$ is a ...
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0answers
28 views

Estimation of trignometric polynomial and lipshitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
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14 views

Every Hilbert Space has an orthonormal basis [duplicate]

I'd be really grateful if someone could tell me what steps I should take (ie. what books to read) before I can prove the statement in the title. I currently have taken rigorous courses in Linear ...
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1answer
29 views

Convergence of a Net: Example

Context I'd like to give my own specific sense to: $$\lim_{t\to 0}U_t=U_0$$ for a unitary group $U_t$. Problem Consider a function $f:\mathbb{R}\to\mathbb{R}$ that satisfies: $$f(t_n)\to c$$ for ...
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0answers
28 views

dual of hilbert space and reisz representation theorem is the same [closed]

i we read in any functional analysis book after discussing dual spaces and proved that the dual of hilbert space H is H the book says the reisz representation theorem my question is why we make ...
2
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2answers
451 views

Prove $g(x)=\sqrt{f(x)}$ is regulated

Let $f:[a,b]→\mathbb{R}$ be regulated and non-negative. Prove that $g:[a,b]→\mathbb{R}$ defined by $g(x)=\sqrt{f(x)}$ is regulated. A function $f:[a,b]\to\Bbb R$ is a regulated function if ...
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1answer
39 views

Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
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1answer
35 views

Identifying a Hilbert space with its dual

Let $H$ be a Hilbert space. Often people say "we identify $H$ with its dual $H^*$ with the Riesz representation theorem". I know there is a map $R:H^* \to H$ which is ismometric and isomorphic. So by ...
2
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1answer
32 views

Cyclic representation on $L^2(\mu)$

Show that if $(X,\Omega,\mu)$ is a $\sigma-$ finite measure space and $H=L^2(\mu)$, then $\pi:L^\infty(\mu)\to B(H)$ defined by $\pi(\phi)=M_\phi$ is a cyclic representation and find all the cyclic ...
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2answers
35 views

Two different results of Fourier Transform $xe^{-x}$

I have a function $f$ defined by $$f(x)=\begin{cases} xe^{-x} \textrm{ if } x>0,\\ 0,\textrm{otherwise}. \end{cases}$$ I wish to know the Fourier transform of $f$, i.e, $${\cal ...
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2answers
27 views

Is the sum of a dense map and a nowhere dense map dense?

Let $X,Y$ be non-zero normed linear spaces, if $A:X\rightarrow Y\ $ has dense image and $B:X\rightarrow Y\ $ has nowhere dense image, does it necessarily imply $A+B\ $ has dense image? $A$ and $B$ are ...
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0answers
20 views

Relation between Compactness and convex hull [closed]

Let (V,∥⋅∥) be a finite-dimensional normed space. Show that, if A⊆V is compact , then so is conv(A).
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1answer
16 views

Show a bounded linear operator is weakly sequentially continuous

Let $X$ and $Y$ be normed linear spaces, $T \in B(X,Y)$, and $\{x_{n}\}_{n=1}^{\infty}\subset X$. If $x_n \rightharpoonup x$, I need to prove that $T x_{n} \rightharpoonup T x$ in $Y$, where ...
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18 views

Relation between cpmpactness and convex hull [closed]

Let $\left ( V , \left \| \cdot \right \| \right )$ be a finite-dimensional normed space. Show that, if $ A \subseteq V$ is compact , then so is $conv(A)$.
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1answer
90 views

The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
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12 views

spectrum of a convolution operator is connected

Let $k \in L^1(\mathbb{R}; \mathbb{C})$ and define a linear operator $T$ acting on $L^2(\mathbb{R}; \mathbb{C})$ by $$ Tf(x) := [k \ast f] (x) \ldotp$$ Linearity is obvious, while the fact that $T$ is ...
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66 views

Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
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3answers
37 views

ONB: Fourier Series

Given the Hilbert space $L^2([-\pi,\pi])$. Consider the orthonormal system: $$\mathcal{S}:=\{\frac{1}{\sqrt{2\pi}}e^{ikx}:k\in\mathbb{Z}\}$$ This is an ONB. How do I prove this? I guess, I could try ...