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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24 views

Definition of $L^2[-\pi,\pi]$ norm.

What is the definition of $$\|f(x)\|_{L^2[-\pi,\pi]}\,?$$ $$\frac{1}{2\pi}\int_{-\pi}^\pi f^2(x)\,dx$$ or $$\sqrt{\frac{1}{2\pi}\int_{-\pi}^\pi f^2(x)\,dx}\,?$$
4
votes
0answers
64 views

Relationship between functional analysis and differential geometry

I am taking courses on functional analysis (through Coursera.com) and differential geometry (textbook author : O'neil) on my university. I made the following table on my own. Are the similar ...
2
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0answers
44 views

Subalgebras of $C_b(X)$ whose elements do not vanish simultaneously at any point

Let $X$ be a completely regular space. How can I find all Banach subalgebras of $C_b(X)$ (all complex-valued bounded continuous functions on $X$) with the property that for every $x\in X$ there ...
6
votes
2answers
50 views

Are the polynomial functions on $S^1$ dense in $C(S^1,ℂ)$?

A friend of mine came up with this problem: Let $S^1$ be the unit circle in $ℂ$ and $P$ the space of polynomial functions $S^1 → ℂ$ (with complex coefficients). Is $P$ dense in $C(S^1,ℂ)$? ...
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0answers
25 views

Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
0
votes
0answers
13 views

Is any quasinormal operator hyponormal?

Is a quasinormal operator hyponormal? In the other words, if $A:H\to H$ is a bounded linear operator such that $A(A^*A)=(A^*A)A $ can we conclude $A^*A\geq AA^*$?
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0answers
20 views

Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
-1
votes
1answer
16 views

The kernel of the dense set [closed]

Suppose the set $S$ is dense in $X$, the operator $T$ is a continuous operator,suppose $Tx\neq 0$ for all $x\in S$, does that imply that $Tx\neq 0$ for all $x\in X$
2
votes
1answer
33 views

Weak Convergence for a specific example

Let $X=(0,1)$ and $u_v(x) = v^{1/p}1_{(0,1/v)}$, where $1 < p < \infty$. It needs to be shown that $\lim_{v \to \infty} \int_X u_v \phi = 0$ for all $\phi \in \mathcal{L}^{p^\prime}$ (The ...
1
vote
1answer
18 views

Show that hermitian element $h=\sum p_n/3^n$ generates $ C_0(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space, and suppose that the C*-algebra $C_0(\Omega)$ is generated by a sequence of projections $(p_n)_{n=1}^{\infty}$. Show that the hermitian element ...
3
votes
2answers
69 views

Problem 11 Section 2.6 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Let $X$ be the vector space of all complex $n \times n$ matrices and define $T \colon X \to X$ by $Tx \colon= bx$, where $b \in X$ is fixed and $bx$ denotes the usual product of matrices. I know that ...
2
votes
1answer
31 views

The derivative of a $L^ {\infty}$ function

If I take the derivative of a function in $L^ {\infty}$ (that is, the function is bounded by a number) in any direction, in which space the derivative is defined? Are there some properties for ...
3
votes
0answers
30 views

Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
3
votes
2answers
30 views

Norm of linear transformation: why restrict ourselves to $\|x\|\leq 1$?

If $f$ is linear transformation from a normed linear space $X$ into a normed linear space $Y$, and define its norm by $$\|f\|=\sup\{\|f(x)\|: x\in X, \ \|x\|\leq 1\}$$ My question is: why restrict ...
1
vote
1answer
45 views

Self-adjointness

In another thread it was claimed that the operator $O : \operatorname{dom}(O) \subset L^2(-1,1) \rightarrow L^2(-1,1)$ is self-adjoint. $$Of(x)= \frac{f(x)}{{1-x^2}}$$ It is obvious that $$\langle O ...
3
votes
2answers
34 views

Difference between an eigenvalue and a spectral value

What is the difference in the definition of a spectral value and an eigenvalue. My notes from functional analysis says $\lambda$ is an eigenvalue of an operator $A$ if $\,\exists \, x \in ...
0
votes
1answer
32 views

If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?

If $f \in L_2(a,b)$, then I want to show that the antiderivative $$ F(x) := \int_a^x f(y) d y $$ is in $L_2$ (I guess this is true). If $L_2(a,b)$ would be closed under pointwise product, i.e. if ...
1
vote
2answers
32 views

Preserve self-adjoint properties

I was thinking about this problem recently: Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f ...
1
vote
1answer
32 views

How to show that the operator $T(\{x_n\})=\{n x_n\}$ has closed graph?

Consider the subspace $$D=\left\{x\in \ell^2 \ \big|\ \sum_{n\in\mathbb N} n^2 |x_n|^2<\infty\right\}$$ of $\ell^2$, and let $T:D\to\ell^2$ be defined by $T(\{x_n\})=\{n x_n\}$. I need ...
0
votes
0answers
12 views

Definition of well-defined for special case

I have a question about what well-defined means in a certain case. For an operator from $X$ to its dual $X^{*}$, say $A:X \rightarrow X^{*}$,why does the definition of $A$ being "well−defined" seem ...
1
vote
1answer
36 views

Fréchet derivatives of $\sum_{n=1}^\infty x_n^2/n^3 -\sum_{n=1}^\infty x_n^4$

I read that the second order Fréchet derivative $F''(0)$ of linear functional $F:\ell_2\to\ell_2$, where $\ell_2$ is the separable real Hilbert space, defined by ...
0
votes
1answer
13 views

The orthogonal operator onto $ran(T)$

I have read that the least square solution for the operator in Hilbert space is given by $$T^*Tx=T^*y$$ where $T$ is the operator $T: X\rightarrow Y$, and $T^*$ is the adjoint operator. Obviously, ...
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votes
0answers
13 views

Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
1
vote
1answer
36 views

Extreme point of unit balls, over $\mathbb C$

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $\mathcal{C}[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
1
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0answers
27 views

Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
2
votes
0answers
31 views

Does nonexpansive mapping imply isometry in this case?

I have the following problem. I want to prove that there exists an isometric isomorphism: $$Lip_0(X) \equiv AE(X)^*$$ Here $(X, d)$ is a metric space, $Lip_0(X)$ is the space (a Banach space with the ...
1
vote
1answer
45 views

Application of Weierstrass' theorem

Consider $f \in C^1[0,1]$ where $C^1[0,1]$ is the space of functions of class $C^1$ on $[0,1]$ furnished with the norm $\|f\| = \|f \|_\infty + \| f' \|_\infty$. Prove that for $\epsilon > 0$, ...
1
vote
1answer
21 views

Holomorphic Functional Calculus vs Borel Functional Calculus

I am currently learning about different kinds of functional calculus and I was wondering if I could get something cleared up. The first type of functional calculus we learned about was holomorphic ...
1
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0answers
35 views

Extreme point of unit balls, the complex case [duplicate]

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $C[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
0
votes
1answer
46 views

Definitions for L2 and Lp Spaces?

I am taking a course in Functional Analysis online, and unfortunately some important terms have not been well defined. In particular, isn't L2 space just Lp space with p=2 ? If so, why aren't ...
1
vote
2answers
68 views

How to prove that trigonometric functions form a Chebyshev system?

How can be proven that $$\{ \operatorname{cos}(kx)\}_{k = 0}^n \text{ and } \{ \operatorname{sin}(kx)\}_{k = 1}^n$$ are Chebyshev systems in the interval $(0, \pi)$? Any ideas will be appreciated. ...
2
votes
1answer
49 views

Book suggestion to prepare the grounds for studying functional Analysis

Hi guys I have 2 month semester break in February and March and I am planning to take a course on functional analysis in 4 months. I have taken a very elementary course on Linear Algebra(Gilbert ...
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0answers
22 views

$A,B$ with $A\subseteq \mathcal{P}(B)$ and $B\subseteq\mathcal{P}(A)$

Given a measurable space $S$, denote by $\mathcal P (S)$ the set of all probability measures over $S$. Do two spaces $A$ and $B$ exist, such that $A\subseteq \mathcal{P}(B)$ and ...
0
votes
0answers
35 views

Derivative of norm in Hilbert space

I read (p. 485 here) that the Fréchet derivative of norm (non-linear) functional $p:H\to\mathbb{R}$, $x\mapsto\|x\|$ is $\frac{x}{\|x\|}$ for all $x\ne 0$, which I think to be intended as the linear ...
0
votes
1answer
28 views

Bump function on set

Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I ...
1
vote
1answer
33 views

How to prove that operators are isometry on $\ell^p$/$c$?

Lately I've been studying Banach spaces and isometries, and encountered many explicit isometrys involving $c$, $c_0$, $c^*$, $c_o^*$, $\ell^1$, $\ell^\infty$, etc... ($c \subset \ell^\infty$ is the ...
7
votes
3answers
88 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
1
vote
0answers
11 views

Strong derivative of a compound map

I find the strong Fréchet derivative of $\Phi(h,\psi(h))$, where $\Phi:T_0\times T_\xi\to Y$ with $T_0, T_\xi, Y$ Banach spaces and $\psi:T_0\to T_\xi$ is strongly differentiable in $0$, evaluated in ...
0
votes
1answer
19 views

Closed restriction of an unbounded self-adjoint operator

Suppose $(A\,;\mathcal{D}(A))$ is an unbounded self-adjoint linear operator (obviously, $\mathcal{D}(A)$ must be dense) on a Hilbert space $\mathcal{H}$. Suppose $\mathcal{D}(C)$ is a proper dense ...
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votes
0answers
8 views

Normalization for argument of maximum function

is it possible to normalize the maximum function of a certain argument ? Means: Is that $\theta_{ML} = arg \max\limits_{\theta} \{ \sum \limits_{n=1}^{N} |w_n w^*_{n+N}| - \Big( \frac{SINR + ...
0
votes
1answer
42 views

Convergence of a sequence in $l_2$

I am wanting to disprove (show that it is not the case) that for a sequence ${x_n} = x^n$, ($n\in\mathbb N$), that if $(x_i)^n \to x_i$ in $\mathbb R$,then $x^n\to x$ in $\ell_2$. I have gotten as far ...
0
votes
1answer
32 views

Solution to functional equation and Cauchy

I want to show that the only continuos solutions to the functional equation $g(x)+g(y)=g(\sqrt{x^2+y^2})$ is $g(x)=cx^2$ where c is a constant i.e. c=g(1). I think that I maybe could rewrite the ...
0
votes
2answers
43 views

A reflexive Banach space is separable iff its dual is separable

Let $(X,||\cdot||)$ be a reflexive Banach space. Prove that $X$ is separable if and only if $X'$ (the dual space of $X$) is separable. Does anyone have a hint for me? I have no idea where to begin
2
votes
0answers
26 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
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votes
2answers
53 views

Is this function defined?

Let a function Let a function $g(f)= \parallel \bigtriangledown f\parallel / sin \parallel f \parallel $ Is $g $ defined for $\left \| f \right \| \leq $ 1? $\left| \left|. \right|\right|$ ...
3
votes
1answer
53 views

Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ ...
0
votes
1answer
29 views

Lower bound and upper bound functions

I am studying for a test and really need to know some examples of function with upper bound and lower bounds. I hope someone can be kind enough to help.Thank you Please give examples of function ...
1
vote
1answer
23 views

Showing $f$ is an $L^p$ function if $f$ is "self-convoluted.

If $f$ is $L^2(\mathbb{R})$ and $f=f*f$, show that $f$ is $L^p$ for $2\leq p\leq \infty$.
1
vote
1answer
73 views

What topics have complex analysis among their prerequisites?

I have one spot left in my bachelor's curriculum and am trying to decide between complex and functional analysis. What the latter is good for, is more or less clear to me: e.g. for advanced ...
2
votes
0answers
33 views

Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...