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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Is the space of bounded linear operators from E (space with an inner product) to C (complex numbers) a Hilbert space?

In other words is there an inner product that produces the operator norm? Let $E$ be a space with an inner product. Show that its topological dual $E^*$ equiped with the operator norm is a Hilbert ...
3
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1answer
48 views
+250

Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta f(v) d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb ...
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0answers
40 views

Show that a complex number's set is no empty

Consider $\alpha \in \mathbb{C}$ such that $Re (\alpha) > |\alpha|^2.$ Why is the set $$\Omega_{\alpha}=\mathbb{C}^{*} - \{\lambda^{\alpha}e^{i\theta\alpha}; \lambda > 0 \ \mbox{and}\ -\pi \leq ...
2
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1answer
59 views

All derivations are directional derivatives [duplicate]

Let $X : C^{\infty}(\mathbb{R}^n) \rightarrow \mathbb{R}$ be a derivation, so i.e. linear and satisfying the Leibniz Rule $$X(fg)=X(f) \cdot g(a)+X(g) \cdot f(a)$$ for some fixed $a \in ...
2
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1answer
22 views

Adjoint operator on Banach space

Suppose $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded linear operator. Show that $T$ is an isometric isomorphism if and only if its adjoint $T^*$ is also an isometric isomorphism. Given an ...
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0answers
9 views

Normal Operators: Backtransform

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$W:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{W}$$ By a ...
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1answer
44 views

Questions about the Kapalansky density theorem

I'm studying Takesaki's Theory of operator algebras book by myself. The following is a theorem from that book: I have several questions about this proof: 1- He claims, in the first line of ...
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17 views

unbounded operator and closed graph theory [on hold]

on this ex we want to prove that the closed graph is not always an graph for linear operator how can i solve this problems any hints this problem is taken from reed-Simon book how can i prove ...
1
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1answer
34 views

Bounded Operator Norm: Special Element

Given a Banach spaces $X$ and $Y$. Consider a bounded operator: $$T:X\to Y:\quad\|T\|<\infty$$ Then theres an element: $$\|Tx\|=\|T\|\cdot\|x\|\quad(x\neq0)$$ Does it always exist?
2
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1answer
24 views

Strong convergence of convex combinations of a weakly convergent sequence

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
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27 views

FUNCTION ANALYSIS [on hold]

if (x,Ω,µ)is a measure space and k ϵl^2(µ×µ),k defines a bounded integral operator.
3
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2answers
399 views

Heat equation and semigroup theory.

Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in X$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function ...
2
votes
1answer
34 views

Sequence in product metric space [on hold]

Let we have $(X_1,d_1)$ is a metric space and $(X_2,d_2)$ is another metric space . Now we will difend $X=X_1*X_2$ and we have $d$ is a distance function on $X$ So $(X,d)$ is a metric Space I ...
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1answer
11 views

Exercise about adjoint of densily defined operator between Banach spaces

Let $X$ and $Y$ be Banach spaces and $A:D(A)\subset X \to Y$ a densily defined linear operator. Suppose the graph of $A$ is closed. Then the follwing are equivalent: $D(A)=X$; $A$ is bounded; ...
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0answers
21 views

Lifting invertible elements in a $C^*$-algebra connected to the identity

Let $A$ and $B$ be unital $C^*$-algebras and suppose that there is a surjective *-homomorphism $f:A\rightarrow B$. Then any invertible element in $B$ that is connected to $1_B$ can be lifted to an ...
-3
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1answer
24 views

Example of sequence function on $C[a,b]$ [closed]

Please give me three examples of sequence $f_n$ in $C[a,b]:= \{ f \colon [a,b] \to \mathbb R \mid f \text{ is continuous} \}$ such that $$ \int_a^b | f_n(x) - f(x)|\ dx \to 0$$ ($L_1$ norm) as $n\to ...
2
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2answers
24 views

An idempotent bounded linear operator has eigenvalues $0,1$

I am thinking of the following problem: suppose $T$ is an idempotent bounded linear operator on a Banach space $X$ over the complex field. Of course, suppose $T$ is not zero map or identity map to ...
0
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1answer
18 views

Showing that the functional $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$ is continuous

Suppose that we have the functional $L: L^2[a,b] \to \mathbb{R}$ , $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$. $f(x)$ is a well behaving, integrable function in $L^2[a,b]$. I want to show that this is a linear ...
3
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2answers
69 views

Eigenvalues and Spectrum

In algebra, I learned that if $\lambda$ is an eigenvalue of a linear operator $T$, I can have \begin{equation} Tx = \lambda x \tag{1} \end{equation} for some $x\neq 0$, which is equivalent to $\lambda ...
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0answers
63 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
2
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0answers
31 views

Projections on a Hilbert space

Suppose $P$ and $Q$ are self-adjoint projections on a Hilbert space such that $P+Q+\lambda I$ is a self-adjoint projection for some $\lambda \in \mathbb{R}$. Does it follow that $P$ and $Q$ commute?
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26 views

Is image of sum of maps closed?

Suppose $V$, $W$ are Banach spaces and $f$, $g:V \rightarrow W$ are continuous linear operators and $\operatorname{Im}f$, $\operatorname{Im}g$ are closed subspaces of $W$. Is it true that ...
2
votes
2answers
47 views

Normal Operators: Draft

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$A:=N(1+N^*N)^{-1}\in\mathcal{B}(\mathcal{H})$$ Then it ...
1
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1answer
27 views

Normal Operators: Transform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$W:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{W}$$ Then it is ...
4
votes
4answers
62 views

$x \perp y$ if and only if $\Vert x + \alpha y \Vert \ge \Vert x \Vert$ for all scalars $\alpha$

Here's Prob. 8 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that in an inner product space, $x \perp y$ if and only if $\Vert x + ...
3
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0answers
37 views

Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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0answers
22 views

Prob. 9, Sec. 3.2 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Here is Prob. 9 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $V$ be the vector space of all continuous complex-valued functions on ...
1
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1answer
58 views

Unitary G-module

I'm not sure if I understand this sentence correctly: "By a unitary G-module we will mean a Hilbert space W on which G acts by means of a strongly continuous unitary representation". $G$ is a ...
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0answers
14 views

About the definition of functional derivative and the $L^2$ inner product

There is something I do not understand well about the definition of the functional derivative. In the wikipedia page http://en.wikipedia.org/wiki/Functional_derivative it says: 1) This definition ...
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0answers
38 views

How can I prove that [on hold]

Let $μ^*$ be an outer measure on $P(X)$ and $m^*$ be the class of measurable sets with respect to the outer measure $μ^*$. How can I prove that the class of measurable set $m^*$ with respect to the ...
3
votes
1answer
83 views

A weak* dense subset intersected with norm ball contains no ball

I'm struggling with this problem in general. Represent $\ell^1$ as the space of all real functions $x$ on $S = \{(m,n): m\geq 1, n \geq 1\}$, such that $$ \|x\|_1 = \sum |x(m,n)| < \infty $$ ...
1
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1answer
48 views

Stone Weierstrass and Runge

Suppose $E(closed)\subset\{z:|z|=1\}$ and let $f(z)$ be a continuous function on the set $E$. I want to show that $f(z)$ can be approximated by polynomials on $E$. I am not exactly sure how to solve ...
1
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1answer
35 views

$L_\infty[0,1]$ completness and separability

Prove that $L_\infty[0,1]$ with the norm given by $$\Vert f\Vert_\infty:= \inf\{S(N): \mu(N)=0\}, \quad \mbox{where} \quad S(N)=\sup\{|f(x)|: x\notin N\}.$$ is complete and is not ...
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2answers
26 views

If $0<r < s$, then $L^s(\mu)\subseteq L^r(\mu)$. Under what conditions do these two spaces contain the same functions? [closed]

Suppose $f$ is a complex measurable function on $X$, $\mu$ is a positive measurable on $X$ and $||f||_p=(\int_X|f|^p d\mu)^{\frac{1}{p}}$ for $0<p<\infty$ If $\mu(X)=1$. $(i)$ $||f||_r\leq ...
-1
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0answers
11 views

Boundary conditions and Lagrange Constraints in Calculus of Variations

I am trying to learn about Calculus of Variations for some time now. In many problems, there are some boundary conditions defined, for example when we want to maximize a functional ...
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0answers
21 views

A dense subset in $L^2(X,\lambda)$

Suppose $S=\{f\in L^2(X,\lambda): f=\alpha_1( \chi_{A_1}-\chi_{X\setminus A_1})+\sum_{i=2}^N \alpha_ i\chi_{A_i}$ where $A_i, A_j$ are disjoint and $a_1>\max_{2\leq i\leq N} \alpha_i \}$. Is ...
1
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0answers
38 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
2
votes
1answer
16 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
130
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0answers
4k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
2
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1answer
28 views

Showing that (on $\mathbb{R}^n$) the $\|\cdot\|_\infty$ norm is weaker than any other norm

Showing that (on $\mathbb{R}^n$) the $\|\cdot\|_\infty$ norm is weaker than any other norm. I am doing past papers and the question is this: "Prove that any norm on $\mathbb{R}^n$ is weaker than the ...
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0answers
10 views

Deficiency indices for differential operator on half-line

1) What is the domain of the adjoint $A^\ast$ of the differential operator $Af = i \frac{d}{dx}$ with $D(A) = \mathcal C^\infty_c (0,\infty)$? 2) I want to compute the deficiency indices of $A$. By ...
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0answers
24 views

About the space of bounded linear operators [closed]

Can you give me any tips on how to show that if B(X,Y) is complete then Y is complete?
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1answer
23 views

$f_n \to f$ in $L^p$, $f_n\to g$ in $L^p{'}$ then $f=g$ a.e $x$.

If $f_n \in L^p\cap L^p{'}$ such that $p\neq p'$ and $f_n \to f$ in $L^p$, $f_n\to g$ in $L^p{'}$ then $f=g$ a.e $x$. a suggestion please.
1
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1answer
12 views

$m_{x_1x_2}\leq m_{x_2x_3}\Rightarrow f$ is convex.

Let $J \subseteq \mathbb{R}$ an interval. If $f:J \to \mathbb{R}$ such that $\forall{x_1,x_2,x_3 \in J}$ $m_{x_1x_2}\leq m_{x_2x_3}\Rightarrow f$ is convex. Where ...
26
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2answers
8k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, ...
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2answers
32 views

Convergence of spectrum with multiplicity under norm convergence

This article by Joachim Weidmann claims that, if a sequence $A_n$ of bounded operators in a Hilbert space converges in norm topology, i.e., $\|A_n - A\| \rightarrow 0$, then "isolated eigenvalues ...
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0answers
14 views

How to calculate full width half max of this curve

I'm trying to calculate the full width half max of this function and I keep receiving non-sense answers. The function is $F(ω)=A_0 (\frac{\frac{1}{τ}}{(\frac{1}{τ^2})+(ω_0 - ω)^2})$ and I then have ...
2
votes
2answers
28 views

A question involving Frechet differentiability

Let $X, Y$ be real normed spaces and $U \subset X$ open subset. In "Nonlinear functional analysis and applications" edited by Louis B. Rall, we have the followint definition (page 115) A map $F : U ...
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0answers
7 views

Lower order perturbations of 2nd order differential operators

Consider the well-known Hormander's sum of squares $P = \sum_{j = 1}^m X_j^2$, where $X_j$ are vector fields on a compact manifold $M$ of dimension $n$. Also assume, as is usual to this theory, $m ...
0
votes
1answer
8 views

Do the elements of a sequence converging to a point in the intrinsic core of a convex cone belong to the intrinsic core of the set eventually?

Let $X$ be a general Banach space and let $C\subset X$ be a convex cone. Consider a sequence $x_n$ in the affine hull of $C$ such that $x_n\to x$ for some $x\in icr(C)$, where $icr(C)$ denotes the ...