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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101 views

What is a concrete example of a non-compact Hermitian operator on an infinite-dimensional Hilbert space whose eigenvectors do not form a complete set?

If I am not misunderstanding anything: by the spectral theorem, Hermitian operators that act upon finite-dimensional Hilbert space as well as compact Hermitian operators that act upon ...
2
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2answers
109 views

Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
3
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2answers
81 views

Solving equations where the solution is an operator

Ok, so here's some context. Solving regular equations we might have something like this: $2 + x = 5$, solving for $x$ we get 3. We might even have an equation like $x + y = 5$ where there are ...
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66 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
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111 views

Linear operator $(Af)(t) = \int_ 0^t e^{t-s}f(s)ds$

Let $X := C[0,1]$ be a Banach space equibed with the norm $||f|| = max_{0 \leq t \leq 1} |f(t)|$. Define a linear operator $A : C[0,1] \rightarrow C[0,1]$ by $(Af)(t) = \int_ 0^t e^{t-s}f(s)ds$. I ...
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3answers
110 views

Norms that $C([0,1])$ to be an incomplete normed space.

I searched all of norms that $C([0,1])$ to be incomplete normed space. But I found only $\|.\|_p$ (for every $1\leq p<\infty$). Are you know another norm on $C([0,1])$ that $C([0,1])$ to be ...
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2answers
199 views

Contraction and Fixed Point

How do I show that for $T: X \rightarrow X$ where X is complete and $T^m$ is a contraction that T has a unique fixed point $x_0 \in X$. I know there exists $\lambda_1 \in (0,1)$ for $x, y \in X$ ...
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1answer
59 views

Either show $c_0=\{x=(x_1,x_2,x_3,…) | x_i\in \mathbb{C}, x_i \rightarrow 0\}$ is complete or closed under supremum norm.

Let $l^\infty =\{x=(x_1,x_2,x_3,...)|x_i \in \mathbb{C}, \|x\|_\infty=\sup_{i\in \mathbb{N}}{|x_i|}<\infty\}$. and $c_0=\{x=(x_1,x_2,x_3,...) | x_i\in \mathbb{C}, x_i \rightarrow 0\}$. So ...
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1answer
46 views

Why must trivial extension of C*-algebra be split Short Exact Sequence?

Background: Suppose $0 \to B \to E -q-> A \to 0$ is a short exact sequence of C*-algebras. Since B sits as an ideal of E, there is a natural *-homomorphism from E to the multiplier algebra of B (by ...
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121 views

exercise on the closed subspaces of an Hilbert spaces

I have a question regarding exercise 3.1.13 of "Analysis Now" by Pedersen volume 118 of the Springer GTM. The exercise aim to show that any closed subspace $X$ of $L^2([0,1])\cap L^{\infty}([0,1])]$ ...
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2answers
169 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
2
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2answers
160 views

Uncountable orthonormal system in Hilbert spaces

I need an example of a Hilbert space in which the following does not hold for all $x$: $$ x=\sum_k^{\infty} \langle x,u_k \rangle u_k. $$ That is, there are elements that are not expressible as ...
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2answers
114 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
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109 views

How would you determine whether this sequence transformation has an inverse?

Let $T : a \mapsto b$ be a transformation of sequence $a$ to $b$ of the form $$ T(a)_m = b_m = \sum_{k=1}^{\infty} a_k e^{-i 2 \pi m / k } $$ Question. How would you go about determining if this ...
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0answers
29 views

Pointwise convergence of resolvent

Suppose $T$ is a quasiniplotent operator and $\lambda_n$ a sequnce converging to $0$. Then clearly $||(\lambda_nI-T)^{-1}||\to\infty$. I am interested for which $x$ we have that ...
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1answer
92 views

Find functions such that under the Cartesian coordinate system $F(x, y) = f(x) g(y)$ but under the polar coordinate system $F(x, y) = h(r)$.

Find all non-constant function $F(x, y)\in C^2(\mathbb{R}^2)$ such that under the Cartesian coordinate system $F(x, y) = f(x)  g(y)$ but under the polar coordinate system $F(x, y) = h(r)$. My ...
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2answers
85 views

The Banach-Steinhaus theorem for seminormed spaces

Assume that we have a vector space $X$ over reals with a countable sequence of seminorms $p_n$ on $X$ such that: $$ p_n(x)\leq p_{n+1}(x) \textrm{ for } n\in \mathbb N, x\in X, $$ $$ \textrm{ for } ...
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1answer
40 views

Inclusions to complete spaces

Let $X$ be any bounded metric space and $B(X,\mathbb{R})$ - a set of all bounded functions $X\rightarrow \mathbb{R}$ which we endow with a norm $||f||=\sup_{t\in X}|f(t)|$. It is easy to verify that ...
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40 views

$A \subseteq (X,d)$ is compact. Which metric $p$ makes $(A \times A,p)$ also compact and $d: (A \times A,p) \rightarrow [0,\infty)$ continuous?

$(X,d)$ is a metric space. And $A \subseteq X$ is a non-empty compact set in the metric space $(X,d)$. Then, does there exists a metrics $p$ and if so which metrics $p$ make $(A \times A,p)$ compact ...
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2answers
146 views

Properties of the duals of $\ell^1$ and $\ell^{\infty}$

a) True or false: (i) $(\ell^{1})^* = \ell^{\infty}$ (ii) $\ell^1 \subset (\ell^\infty)^*$ (iii) $(\ell^\infty)^* \subset \ell^1$ (iv) $(\ell^1)^{**} \subset \ell^1$ b) Give the set of dual vectors ...
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1answer
63 views

Series constructed from a cauchy sequence

Given a cauchy-sequence $\{x_i\}_{i\in \mathbb{N}}$ in a normed space $X.$ I need to construct a series that converges in $\mathbb{R}$ with $\{y_i\}_{i\in \mathbb{N}}$ a sequence in $X$: ...
2
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1answer
139 views

Approximation in Sobolev Spaces

Consider the following proof in Lawrence Evans book 'Partial Differential Equations': How does it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in ...
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1answer
44 views

$C^1$-extension of function on a normal doamin

Let $f(x,t)$ be defined on the set $N:=\{(x,t): x\in(a,b), 0\leq t \leq g(x)\}$ where $g(x)\in C^1([a,b])$, $g>0$ and $f(\in C^1(\bar N))$. Is it possible to extend $f$ smoothly on the set ...
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1answer
47 views

Properties of ON-basis in Hilbert space

Let $H$ be a Hilbert space with an ON-basis $(e_n)_{n=1}^\infty$ and let $A$ be a bounded linear mapping $A:H\to H$ such that $$\sum_{n=1}^\infty\|A(e_n)\|^2<\infty$$ 1: Show that if ...
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1answer
72 views

Is a set of single element $\{x\}$ connected in a metric space $(X,d)$?

Is a set of single element $\{x\}$ connected in a metric space $(X,d)$? Definition: Suppose that $(X,d)$ is a metric space. A set $E \subseteq X$ is said to be disconnected if there exist two ...
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1answer
70 views

Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant. [duplicate]

Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant. I do not know how to do it.
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1answer
154 views

If $a$ and $b$ commute in a $C^*$-algebra and $a$ is normal, then $f(a)$ and $b$ commute for any continuous $f$

I'm trying to find a way to demonstrate the following: Let $(A,*,\|\cdot\|)$ be a unital $C^*$-algebra. If $a,b\in A$ commute and $a\in A$ is normal (i.e. $a^*a=aa^*$), then for every continuous ...
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1answer
83 views

Bounded linear mappings in Hilbert space preserve orthogonality?

My question is the title of this thread! Assume we have a bounded, linear mapping $A:H\to H$ where $(H,\langle\cdot,\cdot\rangle)$ is a Hilbert space, and two non-zero elements that are orthogonal, ...
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35 views

Open maping theorem. Completeness assumption are important [duplicate]

The open maping theorem between banach spaces says. Let $T:X\to Y$ be a linear,continuous and surjective map between the banach spaces $X,Y$ then $T$ is an open map. I need examples to show that the ...
3
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1answer
73 views

Turning an isometric embedding into a homeomorphism

While in studying functional analysis, there is a part of a homework problem from Rudin's Functional Analysis that asks to show that the isometric embedding $\phi: X \rightarrow X^{**}$ is a ...
3
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1answer
158 views

A question on multiplicative linear functional on Banach algebra.

I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume $A$ is a non-unital C*-algebra and $\tilde{A}$ is its unitization (the elements of the form ...
0
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1answer
132 views

does a linear operator has a closed kernel? [duplicate]

Let T be a linear operator from X to Y. If T is bounded, is the kernel of T is closed in X? And, if the kernel of T is closed in X, then is T bounded? thanks . waiting
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110 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where ...
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1answer
150 views

$L_2$ is of first category in $L_1$ (Rudin Excercise 2.4b) [duplicate]

We mean here $L_2$, and $L_1$ the usual Lebesgue spaces on the unit-interval. It is excercise 2.4 from Rudin. There's several ways to show that $L_2$ is nowhere dense in $L_1$. But in (b) they ask to ...
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0answers
106 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
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1answer
52 views

distributional derivative in L^2

Assume I have a function $f \in L^2(R^d,\mu)$ or $f \in L^1(R^d,\mu)$. A. Now I know that it as distributional derivative, right? I call that $\partial f$. B. If I can show now that $\int ...
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1answer
300 views

Whose basis is {1,sin(x),cos(x),sin(2x),cos(2x),…}?

Whenever $f(x)$ is a (Riemann) integrable function on $[-\pi,\pi]$ we can define its Fourier series $f=a_0/2+\sum a_nsin(nx)+b_ncos(nx)$.But we give arbitrary sequence {$a_n$} and {$b_n$},I think ...
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1answer
67 views

proof of lemma 10-8, In functional analysis book of Rudin page 232

In functional analysis book of Rudin page 232, proof of lemma 10-8 We have a function $ h_r(\lambda)= \frac{r^2 g(\lambda)}{z^2(2r-g(\lambda))} , \lambda \in \mathbb{C} $ and $ g(\lambda)$ is an ...
2
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1answer
81 views

Chebyshev polynomials with non-negative constants

Please let help me solve the following problem that I encountered while engaging in my research. I'm dealing with a class of functions, in which each function has a unique series representation of ...
2
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0answers
41 views

Relation between norms on a vector space and norm on a ring

A norm is a special type of metric defined on a vector space. It gives a vector space some topological structure and the hart of Functional Analysis. First time in this morning I was seeing the term ...
13
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1answer
387 views

Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
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1answer
113 views

contraction point?

This is an interesting question I saw in a book online: Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence ...
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105 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
2
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1answer
583 views

Minimum distance between the curves of two inverse functions?

What is the minimum distance between a point on the curve y=e^x and a point on the curve y=ln(x)? What I did: If the curves given were standard one I could find the common normal and solve the ...
1
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1answer
68 views

Do $L^P$ functions form a metric space?

I have a general question about $L^P$ functions. I have heard that $L^P$ functions form a vector space. My question is can we make them form a metric space too? And what is/are the possible metric/s ...
3
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2answers
83 views

Continuous mapping on unit ball such that $T(\Bbb x_0)=0$

got a question from a course in functional analysis. " Let $T:\{\Bbb x\in\Bbb R: ||\Bbb x||\leq 1\}\to\Bbb R^n$ be a continuous mapping. Moreover assume that $\langle T(\Bbb x),\Bbb x\rangle>0$ ...
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0answers
98 views

Fourier transform of displaced airy function

I need to find the fourier transform of displaced airy function.The function is $ψ_n(ξ) = N_n \text{Ai}(ξ − ξ_n)$, where $ξ=x/x_0$, $x_0=(1/2)^{1/3}$, $ξ_n = (3\pi/2)(n − 1/4)^{2/3}$ and $N_n$ is ...
2
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0answers
76 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
3
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1answer
249 views

Compact subset of a Banach space of infinite dimension

Let $X$ be a Banach space of infinite dimension. And let $K\subset M$ be a compact subset of $M$. Can we conclude something about the interior of $M$? It's true that it's empty? I don't know how to ...
2
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1answer
145 views

the solution of Fredholm´s integral equation

Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$. Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f ...