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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Homeomorphism from $K$ to $\Phi_{C(K)}$

Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that ...
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62 views

Problem 2.7.6 in Kreyszig's Introductory Functional Analysis with Applications

Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$. ...
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Spectral Measures: Subspace Decomposition

Attention This thread has been split into this one and: Spectral Measures: Subspace Characterization Problem Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. ...
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30 views

Dynamics: Schwinger-Dyson-Expansion

Given a C*-algebra $\mathcal{A}$ Consider a free generator $\delta_0:\mathcal{D}_0\to\mathcal{A}$ with $\overline{\mathcal{D}_0}=\mathcal{A}$. Introduce a perturbation ...
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14 views

Multivariable calculus of variations (second variation)

So I understand that, if the multivariable Euler-Lagrange equations are satisfied, then the first variation is zero. But that doesn't mean we really are dealing with a minimum of the action or a ...
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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. ...
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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, ...
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28 views

Spectrum of a bounded operator on a (not necessarily Banach) normed vector space

It's well known that on a Banach space, the spectrum of each bounded operator is compact in $\mathbb C$. What about a general normed vector space? Is there a counterexample if we don't assume ...
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41 views

Does the sequence $( n^{1/n} -1)$ belong to any $\ell^p$ space? [closed]

The sequence $( n^{1/n} -1)$ converges to zero but does this sequence belong to the $\ell^p$ space for $p\in\mathbb R$? I don't know the answer, or how one would prove it. Same problem with the ...
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1answer
54 views

Closed subspace. A Hahn–Banach theorem consequence

I am trying to prove: If M is a subspace of a normed space $X$, that $\overline{M}=\bigcap\{\ker(\phi):\phi|_{M} = 0 \}$ It is really easy to see that $\overline{M} \subset ...
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27 views

If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

The following is a remark of Murphy's C*-algebras and operator theory: . I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is ...
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21 views

Lower bound for the norm of the resolvent

I need to prove next statement (I want to do it for general case) $\|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}$ I think it could be like this let $a\in \sigma(A) z ...
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36 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
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33 views

The weak-star topology is Completely Hausdorff (in particular Hausdorff).

Let $X$ be a normed space, $X^*$ its dual space, $(X^*, w^*)$ is completely Hausdorff. Proof: Let $f, g \in X^*$, $f\neq g$ then $\exists x\in X$ such that $f(x)\neq g(x)$ i.e. $\hat{x}(f)\neq ...
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28 views

an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$

Is true that if an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$? Here $\Delta^n$: polydisc and $T^n$: Torus, distinguished boundary of $\Delta^n$.
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48 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
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32 views

Resolvent operator

Let's consider the following operator on $L^2(\mathbb{R}^3)$ $$A(t)=\Delta+b(t,x)\cdot\nabla$$ where $\Delta$ is the Laplace operator and $b(\cdot,\cdot)$ a smooth vector field. How to compute the ...
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149 views

defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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77 views

ideals in $C^*$ algebra

Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$. I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact ...
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44 views

Smallest constant of Lipschitz retraction from bounded to continuous functions

Let $B$ be the space of all bounded functions $f:[0,1]\to\mathbb R$ equipped with the supremum norm*. It contains $C$, the space of continuous functions on $[0,1]$, as a subspace. An $L$-Lipschitz ...
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Meta Euler Lagrange

Consider the following generalized standard Euler Lagrange Problem which is to maximize the quantity $$ \int_{x_1}^{x_2} L(x,y,y', y'' ... y^{(n)}) dx $$ It can be solved by first determining a ...
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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 ...
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29 views

If the dual unit ball of a normed space $X$ is metrizable in the weak-$*$ topology then $X$ is separable

Let $X$ be a normed space and $(B_{X^*},w^*)$ be the unit ball of the dual space $X^*$ endowed with the weak-$*$ topology. Here is a proof a the fact that if $(B_{X^*},w^*)$ is metrizable then $X$ is ...
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32 views

calculate weak derivate of $|x-2|^2$

Let $u$ be a function with $u(x):=|x-2|^2$ on $I:=(-1,1)$. I want to test whether $u \in H^2(I) \backslash H^3(I)$. Let $\phi$ be in $C_0^\infty(I)$. Then: $T_u(\phi '') = \int_{-1}^1 |x-2|^2 ...
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33 views

the norm of a linear operator

In in the demonstration of Lax-Milgramm lemma, they use a linear operator $A:V\to V$, where $V$ is a Hilbert space; My basic problem is how to prove that $$\|Au\|_V=\sup_{v\in ...
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12 views

Functional derivative of $\int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx$ with respect to $f_X(x)$

What is functional derivative of \begin{align*} \int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx \end{align*} with respect to $f_X(x)$. Here $f_{X,Y}(x,y)$ is joint probability density of r.v. $(Y,X)$ and ...
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35 views

Topology induced by semi-norm and P-topology

Let $V$ be an abstract vector space over $F$ and $P$ a family of semi-norms on $V$. We can define a topology on $V$ by declaring its base to consist of subsets of $V$ of the form $\{v\in V | ...
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40 views

Newton method for maps between Banach spaces

I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here): Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a ...
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40 views

Confusion about domains in PDE and “every $C^k$ manifold can be thought of as a smooth manifold”

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like ...
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Showing that the norm of the canonical projection $X\to X/M$ is $1$

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz' lemma and set ...
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35 views

Proving that $X$ is a Banach space iff convergence of $\sum\|x_n\|$ implies convergence of $\sum x_n$

The following is an Exercise of Conway's Functional Analysis. Prove that $X$ is a $\,Banach$ space iff whenever $\{x_n\}$ is a sequence in $X$, such that $\sum \| x_n \| < \infty$, then $\sum ...
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1answer
28 views

Schwartz space on $\mathbb T^{n}$

For the definition of Schwartz space space on $\mathbb R^{n},$ see this. My Questions: (1)Is it make sense to talk of Schwartz space on torus $\mathbb T^{n}$ ? If yes, what can be the analogous ...
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76 views

Showing $f$ is continuous

Let $x_1, x_2, ... , x_n$ be linearly independent vectors of a normed space $X$, so for any $ x \in X$ set $x = \alpha_1 x_1 + \alpha_2 x_2 + ... + \alpha_n x_n$ for real numbers $\alpha ... ...
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38 views

Generators: Analytic Vectors

Given a Banach space $E$. Consider semigroup $T:\mathbb{R}_+\to\mathcal{B}(E)$. Define its derivative: $$A_Tx:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)$$ (The domain being those elements whose limit ...
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1answer
17 views

$\phi(A^+) \subset B^+$ when $\phi: A\to B$ is an isometric linear map

Let $\phi: A\to B$ be an isometric linear map between unital C*-algebras $A$ and $B$ such that $\phi(a^*)=\phi(a)^* (a\in A)$ and $\phi(1)=1$. Show that $\phi(A^+) \subset B^+$. Clearly $A^+ = \{a^*a ...
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30 views

Can a Local Fractional Differential Operator exist?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$. The derivative of $f$ is defined pointwise, and we say that $f$ is differentiable if the derivative exists in each point. Higher order derivatives are ...
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Problem on CR inequality on finite sum [closed]

Let $f$ be a function from {1,2,3,....,10} to R, s. t. $(\sum_{i=1}^{10}|f(i)|/2^i)^2=(\sum_{i=1}^{10} |f(i)|^2)(\sum_{i=1}^{10}1/4^i)$ mark the correct statement. A. there are uncountably ...
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Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
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C*-Algebras: Group vs. Derivation

Given a C*-algebra $\mathcal{A}$. Consider a *-derivation $\delta$. Does it always generate a group: $$\tau(t)=e^{it\delta}$$ But a group of *-automorphisms is a contraction group: ...
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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 ...
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Positive elements of a $C^*$ (MURPHY, ex 2-2).

I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise: Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra. (a) If $a,b$ are positive ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 $$\sqrt{\frac{1}{2\pi}\int_{-\pi}^\pi f^2(x)\,dx}\,?$$
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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 $C(S^1,ℂ)$? ...
3
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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 ...