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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I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $

I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.
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43 views

A simple description of $ {C^{*}}(\Gamma) \otimes_{\sigma} {C^{*}}(\Gamma) $ when $ \Gamma $ is finite.

Problem. Let $ \Gamma $ be a discrete group. Denote its full group $ C^{*} $-algebra by $ {C^{*}}(\Gamma) $. If $ \Gamma $ is a finite group, then is it true that $ {C^{*}}(\Gamma) \odot ...
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132 views

Normal Operators: Spectrum vs. Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
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1answer
35 views

Why is $T:\ell^1\to(\ell^\infty)'$ isometric

The map $T:\ell^1\to (\ell^\infty)', (Tx)(y)=\sum_{n=1}^\infty x_ny_n$ is isometric, but not surjective. According to my book it is easy to prove that $T$ is isometric, but I don't quite know how to ...
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39 views

$T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$

The question goes as follows: $T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$. Given is the data: $X$ is a Hilbert space with an orthonormal ...
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58 views

How to find the spectrum?

Let the Hilbert space $H=l_2$ over the complex field. How to find the point spectrum $\sigma_p(A)$ of $A$ $Ax=(x_1,ix_2,-x_3,-ix_4,x_5,....)$ Any help is very appreciated. thanks :)
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38 views

Completely bounded map and minimal tensor products

Theorem 3.5.2. Let $\phi: A\rightarrow C$ and $\psi: B\rightarrow D$ ($A, B, C, D$ are C*-algebras) be c.p.(completely positive) maps. Then the algebraic tensor product map $$\phi\odot\psi: ...
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36 views

When $A_y$ is invertible?

Given $y\in C[0,1]$ Let $A_y:C[0,1]\rightarrow C[0,1]: x\mapsto xy$ When $A_y$ is invertible? Could you please help.
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61 views

Is the operator $A$ self-adjoint? unitary? normal?

Let the Hilbert space $H=l_2$ over the complex field. Is the operator $A$ self-adjoint? unitary? normal? $Ax=(x_1,ix_2,-x_3,-ix_4,x_5,....)$ could you please help.
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67 views

Monotone property of $L^p$ norm

My question is: Is there monotone property of $\|f\|_p$ when $p$ is increasing, where $\|f\|_p=(\int_a^b f(x)^pdx)^{1/p}$ is the classical $L^p$ norm and $f\in L^p$? . This proposition is ...
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41 views

Using a family of functions to find fourier series

I'm given a family of functions $$T= \left \{\frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt \pi} \cos n\pi, \frac{1}{\sqrt \pi} \sin n \pi: n=1,2,3,\ldots \right \} , $$ on the interval $[-\pi, \pi]$ ...
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135 views

Constructing a sequence that is pointwise bounded but not uniformly bounded by points in a closed, nowhere dense set in $\mathbb{R}$.

I believe that this question below is asking for a sequence of functions that are bounded pointwise in $\mathbb{R}$ but NOT uniformly bounded in a closed, nowhere dense set of $\mathbb{R}$. Suppose ...
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46 views

Characterization of nowhere differentiable functions

Let $N:=\{f\in C([0,1])\vert \text{ f is nowhere differentiable } \}$ and $A_n = \{f\in C([0,1]) \vert \exists x\in [0,1]s.t. \forall y\in[0,1]: |f(x)-f(y)|\leq n |x-y|\}$. Now I have already ...
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80 views

Find the norm of the operator $A:L_2[0,2] \rightarrow L_2[0,2]$ defined by $(Ax)(t) = t \, \operatorname{sgn}(t-1) x(t)$

I have operator: $\boldsymbol{L}_2[0,2] \to \boldsymbol{L}_2[0,2], ( Ax)( t ) = \boldsymbol{t} \operatorname{sgn}(t-1)x(t)$ I need to find operator norm or say that operator isn't bounded. ...
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165 views

Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
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84 views

The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold

Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla ...
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244 views

Prove that weak convergence does not necessarily imply strong convergence without counterexample.

Here is the set of original problems. Let $\{x_n\}$ be a sequence in a normed linear space $X$. Prove that: Strong convergence implies weak convergence with the same limit. The converse ...
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55 views

In the proof of every isomorphism of $\mathbb{C}^n$ onto an $n$-dimensional subspace of a complex topological vector space is a homeomorphism

I was reading the proof of the following theorem in Rudin 2/e: Theorem 1.21 If $n$ is a positive integer and $Y$ is an $n$-dimensional subspace of a complex topological vector space $X$, then ...
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60 views

Compactness, topology

In a general topological space $(X,\tau)$ I have the following situation: $$F\subset M\subset N$$. If I prove that $F$ is compact in $N$ (w.r.t the induced topology), is it true that $F$ is compact ...
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63 views

Why is $L^p$ isomorphic to $(L^p)^2$

Is it possible to say why the spaces in the title are isomorphic as Banach spaces? Is their a Theorem that says this or is it even possible to find an explicit representation of this isomorphism?
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87 views

$C^0([a,b])$ is an infinite dimensional vector space

I am proving that $C^0([a,b])$ is an infinite dimensional vector space. The fact that it is a vector space is clear. But I cannot understand how to prove that it has infinite dimension. Let ...
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155 views

Show that $\log(x)$ is a Bounded Mean Oscillation (BMO)

As an extension of our class notes, we were asked to show that the function $w =\log(x)$ is a Bounded Mean Oscillation (BMO). First off, I believe our professor made a mistake, and really wanted us ...
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81 views

Are the following norms equivalent?

We have the norms $||f||_1=||f||_\infty+||f'||_\infty$ and $||f||_2=|f(a)|+||f'||_\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I prove/disprove this.
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74 views

Multiplication operator has no no eigenvalues

Statement: Let $M_x$ denote the multiplicative operator acting on $L^2([0,1], \, dx)$ by $M_x(f) = xf$. Show that $M_x$ has no eigenvalues Attempt: Let $M_x(f) = xf$ then we should have $M_x(f) = ...
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48 views

weak $L_p$ implies bounded integral on finite measure set

Let $(X, \mu)$ be a measure space which is $\sigma$-finite. $ 1 < p < \infty $. $f : X \to \mathbb C$ is a measurable function. If we know $f$ is in the weak $L_p$ space, i.e. $ ||f||_{L^{(p, ...
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55 views

Prove that $c$ is separable

I need to prove that $c$ - space of all convergent sequences - is seperable. I believe that $c$ is a subspace of $\ell^1$. Now, $\ell^1$ is separable, so $c$ is also separable. Edited: So let $R$ ...
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505 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
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44 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
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38 views

Vector space clarification

I'm asked to decide if the following are vector spaces. A=$\{f:[0,1] \to \mathbb{R}:\int_0^1|f(x)|dx=0$ $\}$ B= $\{f:[0,1] \to \mathbb{R}:f'(x)+4f(x)=0$ and $f(0)=1 $} C=$\{f:[0,1] \to ...
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89 views

Need help understanding this proof about Gelfand spectrum

Consider the following theorem: Let $A$ be a complex non-unital commutative Banach algebra and let $\Omega (A)$ denote its Gelfand spectrum / character space. Then $\Omega (A)$ is locally compact. ...
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41 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
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46 views

Bounded linear operator and self adjoint operator

these questions are in my workbook but there is no worked solutions whatsoever. I dont know where to begin with this at all. Any help would be much appreciated. Thankyou
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47 views

What is the main defferences between nets and ordinary sequences

I know that there are many results in metric spaces (or first-countable topological spaces) can be describe in the language of sequences but these results might not be true in general topological ...
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159 views

Inner product on direct sum of Hilbert spaces

Let $H_1$ and $H_2$ are two different Hilbert spaces then how can we define the inner product on $H_1\oplus H_2$
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77 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
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280 views

What is common between adjoint operator and transpose of the matrix?

I am confused. What is Connection between Adjoint Operator and transpose of the Matrix? I will be very grateful if someone can help me to clarify it.
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107 views

normed linear space of polynomials restricted to $[a, b]$

I have trouble with this problem Let $X$ be the normed linear space of polynomials restricted to $[a, b]$ . For $P \in X$, define $\phi(P)$ to be the sum of the coefficients of $P$. Show that $\phi$ ...
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253 views

Borel functional calculus

For a normal operator T, we have a resolution of the identity $\int_{{\sigma}(T)} {\lambda}\,dE=T$. If $T$ is in addition compact , we have that $\sum_{n=1}^{{\infty}}{\lambda}_{n}\langle ...
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130 views

Prove that — the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$

Prove that the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$.
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1k views

Spectral Mapping Theorem

Spectral mapping theorem is as follows: https://math.uc.edu/~halpern/Matrix.methods/Homatrixmethods/Spectralmappingthm.pdf Is Spectral mapping theorem true for point spectrum ?
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57 views

A proposition about cyclic representation in C*-algebra

Let $A$ be a C*-algebra, if for an arbitrary cyclic representation $\pi: A \rightarrow B(H)$, we have $\pi(a) \geq 0$, $a\in A$, then can we conclude that $a \geq 0$?
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157 views

A theorem about a tracial state in von Neumann algebra

I am reading a book about C*-algebra. There is a quotation below. Let $M$ be a von Neumann algebra with a faithful normal tracial state $\tau$ and let $1_{M}\in N\subset M$ be von Neumann subalgebra. ...
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68 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
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58 views

Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?
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121 views

How to show that the monomials are not a Schauder basis for $C[0,1]$

why the monomials are not a Schauder basis for $C[0,1]$? $p_n(x)=x^n$ such that $(p_n)$ does not form a Schauder basis for $C[0,1]$ span$\lbrace p_n : n\ge 0\rbrace$ is dense in $C[0,1]$ by ...
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125 views

compactness in the space of analytic functions

I am always getting confused by the idea of compactness so I would like some help to see whether a set is compact. (I need this to prove existence of a solution of a map) So let $D\in\mathbb{C}$ be ...
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61 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
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46 views

Non-empty closed subset of the complex plane is the spectrum of a normal operator

This is an exercise in Chapter 13 of Rudin's Functional Analysis. The question is to show that every non-empty closed subset of $\mathbb{C}$ is the spectrum of some normal (not bounded) operator in a ...
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57 views

When sup norm,i.e. $\| x|| = \sup|x(t)|$ for $\forall t\in T $in C[0,1] for $T \subsetneq [0,1]$

When sup norm, i.e. $\| x|| =\sup|x(t)|$ for $\forall t\in T $in C[0,1] $T$ is such that $T \subsetneq [0,1]$. What condition should be applied to $T$ to make $\| x||$ a norm. I cannot show for ...
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157 views

$c_0$ is a closed subspace of $l^{\infty}$

Put $$ l^{\infty} = \{ (x_n) \subseteq \mathbb{C} : \forall j \; \;\ \;|x_j| \leq C(x)\} $$ I want to show that $c_0$, the space of all sequences of scalars that converges to $0$ is closed subspace ...