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, ...

learn more… | top users | synonyms (1)

1
vote
1answer
45 views

Hilbert space and Parallelogram law

Let $ C_\infty$ be inner product space of all real sequences $\{x_n\}$ with $x_n$ finite number of nonzero terms and the inner product defined by $$\langle x,y\rangle =\sum_{i=0}^\infty x_ny_n$$ I ...
2
votes
0answers
38 views

Spaces of the derivative in a direction

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are. Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional: $$\phi =L^p ...
7
votes
1answer
19 views

nonempty open set in normed space is connected iff each pair of points of the set can be joined by a polygon that lies wholly in the set

Let $E$ be a normed vector space. Let $x_1, \dots, x_m$ be points of $E$. Let $f(t) = (k-t)x_k + (t - k + 1) x_{k+1}$ for $k-1 \le t \le k$, $k = 1, 2, \dots, m-1$. The set $\{f(t)\text{ }|\text{ }0 ...
3
votes
1answer
23 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
0
votes
2answers
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
votes
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 $C(S^1,ℂ)$? ...
0
votes
0answers
24 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 $A^*A\geq AA^*$?
1
vote
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
15 views

The kernel of the dense set [on hold]

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
32 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
17 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
68 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 ...
0
votes
0answers
23 views

Banach Algebra spectral theory [closed]

Let $(\Omega, \mu)$ be a measure space. Show that the linear span of the idempotents is dense in $L_\infty(\Omega, \mu)$.
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
11 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, ...
0
votes
0answers
85 views

If every $M\subset X$ closed is such that $M^{\bot\bot}=M$, then $X$ is Hilbert space. [closed]

If $X$ is an inner product space and if $M^{\bot\bot}=M$ for every closed subspace $M$ of $X$, then $X$ is a Hilbert Space. Can someone help-me?
0
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
vote
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
30 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
vote
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
43 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. ...
1
vote
1answer
47 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 ...
0
votes
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
32 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
87 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 ...
0
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 ...