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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31 views

Do I have to show this map is well-defined?

Let $H$ be a Hilbert space and $u \in B(H)$. Write $$ H = \overline{\mathrm{im}(u)} \oplus \overline{\mathrm{im}(u)}^\bot$$ and define $v(h) = v(|u|x \oplus z):= u(x)$. Do I have to prove that ...
1
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1answer
90 views

Derivative of an L1 norm of transform of a vector.

I have to take derivative of the l-1 norm. L1 is the function R in the following expression: $$ R(\psi Fx) $$ where x is a vector, F is the inverse Fourier transform, and $\psi$ is a wavelet ...
1
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1answer
56 views

Another property of the function $\phi : p \mapsto \int |f|^p d\mu$.

Let $(X,\mu)$ be a finite measure space and $f:X \to \mathbb{R}$ a real valued measurable function. Define $E=\{p : \int |f|^p <\infty\}$ and $\phi : E \to [0,\infty]$ by $\phi: p \mapsto \int ...
2
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1answer
210 views

Exchanging max and limit

Suppose I have sequence of function $f_n$ that converge to $f$. Suppose I want to find maximum of $f$ over some set $S$ that is \begin{align*} x^*={\rm arg} \max_{x \in S} f(x) \end{align*} ...
2
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1answer
64 views

Injective tensor product

We know that $c_0\check\otimes c_0=c_0(c_0)$ where $\check\otimes$ is the injective tensor product. is the following still true? $$c_0\check\otimes l^\infty = l^\infty(c_0).$$ Thank you for your help ...
3
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2answers
84 views

Approximate Identities

Statement : Let $U$ be a C* algebra and$\lambda=\left\{A\in U:A\geq 0, ||A||<1\right\}$. If $B\in \lambda$ then if $X\in U$ $$||X^*(I-B)^2X||\leq ||X^*(I-B)X||$$ For reference this is from ...
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1answer
60 views

A proof of theorem in functional analysis

So, the theorem states the following: Let $f:(a,b)\to \Bbb R$ be a function. Then $f$ is convex if and only if it holds: $(1)\quad(\forall x,y,z) (a<x<y<z<b),\left| \begin{array}{ccc} 1 ...
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2answers
39 views

Prove that this function is limited from the down side and not from the upper side.

$y=\frac{12}{x} \quad \text{and} \quad x >0 $Well it shows a part of a hyperbole and $x$ shouldn't be $0$.But how can I prove that is is limited from the down side? EDIT: Please no limits
2
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1answer
52 views

Application of Fubini Theorem in Quantum Mechanics

I'm afraid I'm very confused by how to correctly apply the Fubini theorem to simplify integrals? I have some integral $$ \sum_{k = 0}^{2}\int_{0}^{T} dt_2 \int_{0}^{t_1} dt_1 \bigg[ ...
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1answer
68 views

Spectrum of an upper triangular infinite matrix

Is there any theory about spectra of triangular infinite marices? If so, where could I read something about it?
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50 views

Showing a Set is Dense in $C[a,b]$

Show that $C_0[a,b]:= \{f \in C[a,b] : f(a)=f(b)=0 \}$ is $\| \cdot \|_2$-dense in $C[a,b]$. Is it also $\| \cdot \|_{\infty}$-dense in $C[a,b]$? In the relevant chapter of my textbook, I am given ...
2
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1answer
39 views

Nash inequality : does $f\in L^1$ and $\nabla f \in L^2$ implies $f\in L^2$?

Let $f$ be any function that belongs to $L^1(\textbf{R}^d)\cap H^1(\textbf{R}^d)$ ($d$ a positive integer). Nash inequality applies in this case and gives us $$\| f\|_{L^2}\leq C \| f\|_{L^1}^r \| ...
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0answers
25 views

Discrete Laplace: ONB

Before, consider the discrete Laplace without boundary: $$\Delta:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(\Delta u)_k:=\frac12(u_{k-1}+u_{k+1})$$ Regard the unitary transformation: ...
1
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1answer
42 views

Compactness in Infinite dimension

I was looking for a characterization of the dimension of an nvs using Heine Borel theorem. suppose i have a Compact operator between an Hilbert space and itself, i want to proof that the autospace ...
2
votes
2answers
49 views

Properties of the multiplication operator, self-ajointness

Let $(\Omega, \Sigma,\mu)$ a measurable space, $f:\Omega\to \mathbb{R}$ $\mu$-measurable. a.My first question: What does "f $\mu$-measurable" mean?I only know, what it means that "f is measurable" but ...
0
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1answer
26 views

Continuity of operators

Let $(T_t)_{t \ge0}$ be a family of operators(not necessarily bounded, but all defined on the same domain) and now we have the property $$t \rightarrow 0^+ \Rightarrow ||T_t^2 -T_0^2|| \rightarrow ...
3
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1answer
67 views

Classes of measures that are closed under multiplication

Consider the space $\mathcal M$ of all finite complex Borel measures on a segment with norm $\|\mu\|=\int d\,|\mu|$. Assume that a norm-closed linear subspace $\mathcal M_0$ of $\mathcal M$ has the ...
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2answers
99 views

Square root of the operator $T$

Find the positive square root of the operator $T$ on $L^2 ([a,b])$ defined by $(Tf)(t) =g(t)f(t)$, where $g$ is a positive continuous function on $[a, b]$.
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2answers
41 views

$L^1$ limit of indicator functions must be an indicator

I'm trying to solve this question from Folland's book: Assume $\mu(E_n)<\infty$ and $\chi_{E_n}\to f$ in $L^1$, so $f=\chi_E$ a.e, for some $E$. I don't have much clue.. I think that $E$ should be ...
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1answer
36 views

For $f \in L^1[0,1]$ such that $\int_0^1 x^nf(x)dx = 0$ $\forall n \in \mathbb{N}_0$, then $f(x) = 0$ almost everywhere.

Prove that if $f \in L^1[0,1]$ is such a function such that for all $n = 0,1,2,\dotsc$, $$\int_0^1 x^n f(x) \, dx = 0,$$ then $f(x) = 0$ almost everywhere. Intuitively, I see this to be ...
3
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2answers
120 views

Are these linear maps bounded?

Let $\mathcal{C}^{\infty}_c$ be the complex vector space of $\mathcal{C}^{\infty}$ functions with compact support in $(0,1)$.Define two norms on it , $\|x(t)\|_u=\text{max}_{t\in (0,1)} \ |x(t)|$ and ...
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1answer
39 views

Clarification needed for closed sets

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is closed in $BC(\Bbb R, \Bbb R)$ but not closed in $C([−a, a], \Bbb R)$ for ...
3
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0answers
126 views

Cadlag function, metric

Please to meet you. I have a question about a càdlàg function and its space. Ma & Röckner's book (page 93) contains the following metric space $(\mathcal{D},d)$. Let $E$ be a complete separable ...
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1answer
75 views

Stone-Weierstrass: Summary

This is just a summary. Theorem Given a compact domain $\Omega$. Regard the function space: ...
2
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1answer
85 views

A consequence of the open mapping theorem

We let $f$ be a bounded and surjective linear map from the Banach space $X$ onto the Banach space $Y$ and put $$ r_0=\inf\{r: f(B^X(0,r)\supset B^Y(0,1)\}. $$ Using the open mapping theorem, I have ...
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1answer
38 views

Stone-Weierstrass: Lattice

This is just a prework. Given a compact domain. Regard the function space: $$\mathcal{C}(\Omega,\mathbb{R}):=\{f:\Omega\to\mathbb{R}:f\text{ continuous}\}$$ Clearly it is an algebra: ...
2
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0answers
61 views

question about density of Sobolev spaces

I have a short question about density of spaces. Consider: $C_c^{\infty}(0,1)=\{f\in C^{\infty}(0,1); supp(f)\subset (0,1)\;\text{compact}\}, $ ...
4
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2answers
274 views

Arzela-Ascoli: Proof?

Problem Given a compact domain. Regard the function space: $$\mathcal{C}(\Omega):=\{f:\Omega\to\mathbb{C}:f\text{ continuous}\}$$ Consider a bounded family: ...
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1answer
85 views

A question about complement of a closed subspace of a Banach space

Let $X$ be a Banach space and $M$ be a closed subspace of $X$. Suppose that there exists a subspace $N$ of $X$ such that $X=M\oplus N$. Does it imply that $N$ is closed ? I know that not every ...
3
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1answer
35 views

Approximating $f \in L^1(M)$ with integral zero with $f_n \in L^\infty(M)$ with integral zero?

Let $M$ be a closed Riemannian manifold and let $f \in L^1(M)$ with $\int_M f =0$. Is it possible to find a sequence $f_n \in L^\infty(M)$ with $\int_M f_n = 0$ such that $f_n \to f$ in $L^1(M)$? I ...
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1answer
60 views

Compact, bounded , closed.

Let be X a Normed Vector Space, my question is: if a set A contained in X is compact, bounded, closed, is it finite dimension? I was looking for a characterization of the dimension of an nvs using ...
0
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1answer
46 views

Stone-Weierstrass: Literature

Short question... Does someone know some textbook, a paper or notes that treats: An algebra of functions with identity that separates is dense within a function space.
1
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1answer
48 views

Show that $\mathcal U$ is a contraction mapping

I have $$ \mathcal U:\mathcal C([0,T],L^p(\mathbb R^d)) \rightarrow \mathcal C([0,T],L^p(\mathbb R^d)) $$ Linear such that $$ \frac{d}{dt}||\mathcal U(h)(t)||_p^p\leq C_1\,||\mathcal U(h)(t)||_p^p + ...
2
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1answer
84 views

Inverse transform of a modified Abel transform

I have been struggling for 6 months on finding the analytical inverse transform of a transformation below: $$F(y,k) = 2 \int_y^{\infty}\cos\left(ka\sqrt{r^2-y^2}\right) f(r,k) ...
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0answers
46 views

null power element in a C*-algebra

Let $A$ be a C*-algebra. Show that there is $x\in A$ such that $x^2=0$. I think in abelian C*-algebra $x^2=0$ if and only if $x=0$(because these elements are continuous functions) Also in certain ...
9
votes
1answer
149 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
3
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1answer
39 views

Linear independence of linear functionals on function space

I would like to show or verfiy the following Here is my setting. Let $F$ be a finite dimensional function space over the real $R$. Let $L_1, L_2,\ldots,L_n$ be continuous linear functionals on $F$. ...
1
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1answer
26 views

Uniqueness of continuous linear extensions to the second dual

Suppose that $X$ and $Y$ are normed vector spaces (not necessarily Banach or reflexive) and $T:X\to Y$ is a continuous linear functional between them. Consider the functional $T'':X''\to Y''$, mapping ...
4
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0answers
101 views

What is the relationship beteen functional analysis and topology

Could someone explain in layman's way using examples how is topology related to functional analysis? (edit) After taking a UG course in Point-set topology i felt to have a taste of functional ...
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2answers
25 views

Exponent of an operator - Existence/Uniqueness?

I have the following questions: When I can define an Expression $A^p$ with an Operator $A$ and a fractional Exponent $p$? Is the root (or fractional or even real exponent) existing for arbitrary ...
2
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1answer
90 views

Multiplication Operator and Supremum Norm

Let $m\in C[a,b]$. Consider on $(C[a,b], \|\cdot \|_{\infty})$ the multiplication operator $A: C[a,b] \to C[a,b], \quad Af = mf$. Prove that $\|A\| = \|m\|_{\infty}$. In my book, we are given the ...
1
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1answer
49 views

Linear Operator bounded on a basis

Given a Hilbert space $\mathcal H$, a basis $\{e_j\}$ and an injective function $T$ from $\{e_j\}$ to $\mathcal H$ such that $\| T(e_j) \| \leq C$ for all $j$. Can we always extend $T$ to a bounded ...
2
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1answer
60 views

positive element in a Banach $*$- algebra

By definition, $a$ is positive in C*-algebra $A$ if $\sigma(a) \subset \Bbb R^+$. I would like to know the definition of a positive element in a Banach $*$-algebra. I think it's the same as the ...
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1answer
35 views

Checking if a matrix defines a bounded operator

Does this matrix define a bounded operator from $\ell^2$ to $\ell^2$? \begin{pmatrix} 0 & 1 & \frac { 1 }{ 2 } & \frac { 1 }{ 3 } & ... \\ -1 & 0 & 1 & \frac { 1 }{ 2 } ...
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1answer
49 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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0answers
41 views

Doubt about convergence of a sequence in $H^1(\mathbb{R}^3)$

Let's consider a sequence $\{f_n\}_n$ of $C^\infty_0(\mathbb{R}^3)$ complex-valued functions and suppose thet $f_n\to f$ strongly in $H^1(\mathbb{R}^3)$. What can I say about the convergence of the ...
0
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2answers
49 views

Non-linear analysis

1) I am looking for a book which would give the proof of the following theorem(see below). I didn't find any book who does it: in infinite dimension (in Rockafellar Convex analysis book we are in ...
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2answers
90 views

Trouble Applying Hahn-Banach Theorem

The Hahn-Banach Theorem is described as this: $X$ be a real vector space and $p$ a sublinear functional on $X$. Furthermore, let $f$ be a linear functional which is defined on a subspace $Z$ of $X$ ...
2
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0answers
30 views

Preimage of Legendre-Fenchel transform

Let $X$ be a Banach space with dual $X'$, and let $f : X'\to (-\infty,+\infty]$ be a convex lower semicontinuous function. Does there exist some characterization or some nontrivial results concerning ...
2
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1answer
60 views

Is this proof of Schur's lemma (for a densely defined closed operator) mistaken? How to fix it?

I'm trying to understand a version of Schur's lemma for a densely defined closed operator. It is on page 17 of the book Nonabelian Harmonic Analysis by Howe and Tan. The confusing parts are ...