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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14 views

Prove that every reproducing kernel is a positive matrix (and vice versa)

Let $\mathcal{H}$ be a functional hilbert space (defined over a set $S$) with a reproducing kernel K. Prove that: a) $K$ is a positive matrix means the queadtric form is positive, i.e ...
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24 views

Extension of measure beyond Jordan-measurable sets

I know that if a set $A$ is Jordan-measurable (according to the definition that can be found here in problem 8) with respect to measure $\mu$, then, for any measure $\tilde{\mu}$ that is an extension ...
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24 views

Extension of $\sigma$-additive measure beyond Lebesgue-measurable sets.

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined ...
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0answers
29 views

Parabolic holder norms

Let $Q=\Omega\times[0,T]$ be a cylinder with $\Omega$ bounded open set in $\mathbb{R}^N$. N.V.Krylov in "lectures on elliptic and parabolic equations in holder spaces" defines the parabolic holder ...
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0answers
10 views

A basic doubt on metrizable topological vector space [duplicate]

Suppose we have a topological vector space with a countable local base. Then it has a balanced $\{V_n\}$ such that $$V_{n+1} + V_{n+1} + V_{n+1} + V_{n+1} \subset V_n$$ Why ? I know that every ...
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0answers
30 views

Approximate unit of a separable C*-algebra

The following is a corollary of Takesaki's Operator Theory: My question: I do not know why the author says"there exists an n such that $||x(1-v_n)^\frac{1}{2}||<\epsilon$" . Please help me to ...
8
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2answers
89 views

Why are $L^p$ spaces for $p\not=1,2,\infty$ important?

$L^p$ spaces for arbitrary $1\le p\le\infty$ are a mainstay of basic functional analysis courses, but I've only seen them "in action" when $p$ is 1, 2, or $\infty$. Can anyone give an "elementary" ...
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1answer
29 views

Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence

Let $\{e_n\}$ be a complete orthonormal sequence in an Hilbert space $H$ and let $\{f_n\}$ be an arbitrary sequence of elements in $H$ s.t $$\sum_{n=1}^\infty\|f_n-e_n\|^2<1$$Show that ...
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4answers
76 views

What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
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1answer
20 views

Proving for each seperatble hilbert space exist complete sequence

Let $H$ be a separable Hilbert Space. Prove that exists orthonormal complete sequence and give example for one non-orthonormal sequence. I thought taking orthonormal basis for $H$ denoted by ...
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23 views

Equality of extensions of Jordan measure

I find the following theorem in Kolmogorov-Fomin's Elements of the Theory of Functions and Functional Analysis (p. 280 of this Russian ed., p. 26 of 1963 Graylock English ed.): In order that two ...
5
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1answer
104 views

Is $\mathcal{B}(H)$ complemented in $\ell_\infty(I, H)$

Let $H$ be an infinite diensional Hilbert space. Consider unit ball of $H$ as index set, denote it by $I$, then we have an isometric embedding $$ ...
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1answer
42 views

Possible inconsistency in a subspace of $l^∞$

Is it possible to have a suspace of $l^∞$ in which every sequence has a finite number non-zero elements? if so, what would be the zero element of the space? This a problem of the book of Kreyszig ...
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1answer
31 views

An routine exercise about matrix norm

If $T_{n}\in M_{k(n)}(\mathbb{C})$ and $||T_{n}^{*}T_{n}-1_{k(n)}||\rightarrow0$, then $||T_{n}T_{n}^{*}-1_{k(n)}||\rightarrow 0$ too? (Here, $M_{k(n)}(\mathbb{C})$ denotes the $k(n) \times k(n)$ ...
3
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29 views

Norm of an element in a C*-algebra

The following is a part of a proof in Takasaki's Operator theory: Let $\epsilon>0$ and $A$ is a C*-algebra. For an $x\in A$, put $h=x^*x$ and $u_\epsilon=(h+\epsilon)^{-1}h$. We have then ...
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2answers
52 views

Sobolev space on union of two open sets

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and ...
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1answer
21 views

Finite dimensional subspaces are always closed

If $n$ is a positive integer and $Y$ is an $n$-dimensional sub-space of a complex topological vector space $X$ then I have to prove that $Y$ is closed. Proof: Let $S$ be the sphere which is the ...
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1answer
18 views

Does a Reproducing Kernel Hilbert Space of functions always have a distance defined in it?

Recall that a (reproducing kernel hilbert space) RKHS has two equivalent definitions: 1) Its a Hilbert space of functions $\mathcal{H}$ (i.e. vector space with an inner product $\langle \cdot, \cdot ...
2
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1answer
49 views

What is the image of operator exponential?

Given a Banach space $V$ and a bounded linear operator $A:V\to V$, the operator $e^A$ is bounded and invertible. When $V$ is finite dimensional, every invertible operator is of the form $e^B$ (one can ...
3
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1answer
73 views

Cardinality of basis of endormophism algebra

Is there a relation between the cardinality of the basis of a vector space $V$ over $k$ and the cardinality of the basis of $\operatorname{End}{V}$, the set of $k$-linear endormophisms of $V$, over ...
2
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1answer
32 views

$\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$

I want to prove this $\|T^*T\|=\max\{\lambda: \lambda \text{ is an eigen value of } T^*T\}$\ Suppose $\lambda_0>\lambda_1>\dots>\lambda_{n-1}\ge 0$ are distinct eigen values of $T^*T$ and ...
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0answers
16 views

Show that $H=\oplus_{\alpha \in \sigma(T)}\text{ker}(\alpha I -T)$

Suppose $T$ is an operator on a Hilbert space $H$ such that $\sigma(T)=\sigma_{e}(T)$, and for each $\alpha \in \sigma(T)$, the corresponding eigenspace ker$(\alpha I-T)$ is a reducing subspace for ...
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1answer
15 views

Is there exists other statements equivalent to the analytic rank?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The order of vanishing (the analytic rank) at a point $s=a$ is denoted by $m$ (the ...
2
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34 views

When $\int_\Omega f(x)K(x,y)dx$ is injective

Let $T:X\to Y$ be a compact integral operator and consider $g=Tf$, i.e. $$g(y) = \int_\Omega f(x)K(x,y)dx$$ where $\Omega=[a,b]$ is some subset of $\mathbb{R}$. Are there some particular conditions on ...
11
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1answer
118 views

Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?

It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
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1answer
28 views

Spectrum of a Multiplication Operator

I have a question involving the spectrum of a multiplication operator. We are in the space of square integrable functions over $\mathbb{R}$, $L^2(\mathbb{R})$, and we define $$(T\psi)(x)=f(x)\psi(x)$$ ...
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0answers
35 views

Definition and Intuition of a Weakly Dense Set

What does it mean to say: set A is "weakly dense" in a set B? The definition of a "dense set" is rather intuitive: the classic example of Q (rationals) being dense in R (reals) is very clear. How ...
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38 views

$\sigma$-additivity of an abstract measure

I know and have been able to prove the following lemma: Let $X$ be a set and $\mathfrak{M}$ a $\delta$-ring of subsets of $X$. The set $A\subset X$ is defined as measurable with respect to ...
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30 views

Tensor products, help with proof

Let $X$, $Y$ and $Z$ be Banach spaces. Let the space $X\otimes_{\epsilon}Y\otimes_{\epsilon}Z^{*}$ be the injective tensor product. The injective norm is defined as follows: ...
3
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29 views

The positive element in a C*-algebra

The following is a theorem of Conway's Functional Analysis: for the proof ($c\to a$), I think we can say: for $\lambda\in \sigma(a)\subset \Bbb R$, there is a character $h:C(\sigma(a))\to\Bbb C$ ...
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1answer
32 views

Prehilbertspace = Inner Product Space = Unitary Vector Space?

Are pre-hilbert space, inner product space, unitary (in the complex case) or euclidian (in the real case) vector space the same things, just synonyms?
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23 views

Existance of the integral in the domain of generator of the strongly continuous semigroup

Let $\{s(t)\}_{t\geq 0}$ is a $C_0$ semigroup of bounded operator on the Banach space $X$ and $A:D(A)\subset X\rightarrow X$ be the infinitesimal generators of the semigroup $\{s(t)\}_{t\geq 0}$. ...
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2answers
36 views

Self-adjoint elements in a C*-algebra

I have a simple question which confused me. Suppose $A$ is a C*-algebra. every $x\in A$ has a representation such as $x=a+ib$ where $a,b$ are self-adjoint elements of $A$. Also we claim that $x^*x$ ...
8
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1answer
98 views

Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \begin{equation} \|f\|_{wL^p}=\sup_{\gamma>0} ...
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1answer
25 views

A basic doubt on topological vector space

Consider a topological vector space and a balanced neighbouhood of $0$ in it. Assume that it is not bounded. Is it true that it is the whole space ? why ?
2
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0answers
28 views

*-isomorphism of a C*-algebra into an involutive Banach algebra is norm increasing

The following is a proposition of Takesaki's Operator theory: My question: How does he assume, considering the C*-subalgebra generated by k instead of $*$-Banach algebra B? Are we sure that the ...
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1answer
34 views

when norm of an operator is given by max of eigen values modulas

Could any one tell me how this $\|x\|^2=\|x*x\|$ and the rest of it? I know $\|x\|=\|x^*\|$, I also understand $x^*x$ is hermitian and so diagonalizale but then did not understand the norm square ...
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1answer
47 views

Prove $\dim X = \mathfrak{c}$ for every infinite dimentional Banach space

Let $X$ be an infinite-dimensional Separable Banach Space. Prove that $\dim X=\mathfrak{c}$. On the direction of $\dim X\ge \mathfrak{c}$, I thought taking the subset of all elements $x\in X$ ...
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20 views

Reference: proof of Cramer-Rao

I'm looking for a detailed reference of dealing with the proof of the multivariate case of Cramer-Rao lower bound.
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1answer
27 views

Operator on the space of square summable sequences

We define an operator $T:\mathcal{l}^2(\mathbb{Z})\rightarrow\mathcal{l}^2(\mathbb{Z})$ where $\mathcal{l}^2(\mathbb{Z})$ is the Hilbert space of square summable functions, such that for ...
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4answers
101 views

how to interpret this norm

Could any one tell me what "x" is there when he has defined $\|x\|$, just after he says $M_n$ has the operator norm thank you for helping.
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0answers
24 views

Constructing frame for $C^2$ such that frame coefficients are not unique.

Theorem: Let {$f_k$}$_{k=1}^m $ be a frame for a finite-dimentional vector space $V$. Given $f \in V$, there exists coefficients {$d_k$}$_{k=1}^m \in C^m$ where $C$ set of complex numbers, such that ...
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1answer
43 views

An example where the supremum of Riesz's Lemma is not achieved

The Riesz's Lemma says, if $X$ is a Banach Space with norm $\|\cdot\|$ and $L$ is a closed subspace of $X$, then we have $$ \sup_{f:\|f\|=1} dist(f,L)=1, $$ where $dist(f,L)=\inf_{g \in L} \|f-g\|$. ...
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0answers
26 views

Fourier Series Operation Is A Linear Operator

I am sort of stuck on this problem. Here it is: Show that the Fourier Series Operation is Linear, that is, show that the Fourier Series of $c_1f(x) +c_2g(x)$ is the sum of $c_1$ times the Fourier ...
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1answer
20 views

Compact Operator Inversion

Let I be a positive compact in $\mathscr{B}(\mathscr{H})$ (where $\mathscr{H}$ is some Hilbert space) then $I$ can be written (uniquely) as $A^2=I$ for some $A \in \mathscr{B}(\mathscr{H})$. My ...
1
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1answer
39 views

Does the supremum is finite?

Let $B(A)$ be the space of all bounded functions on a given set $A$, define a metric as follows $$d(x,y)=\sup \{|x(t)-y(t)| : t \in A \}.$$ Show that the supremum exists ?
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50 views

Infinite Holder inequality

Let $\{p_n\}$ be a sequence of real numbers in $[1,\infty]$ such that the sum $$\sum_{n=1}^\infty \frac{1}{p_n} = \frac{1}{r}.$$ Let $\{f_n\in L^{p_n}(\Omega)\}$ be a sequence of functions such that ...
0
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2answers
40 views

How is this boundary condition $\lim_{t \rightarrow -\infty} f(x,t) = y_0(x)$ for a PDE called?

If you have a diffusion equation $\partial_t f(x,t) = \partial_x^2 f(x,t) $, where $(x,t) \in [0,a] \times \mathbb{R}$ and then you say $\lim_{t \rightarrow -\infty} f(x,t) = y_0(x)$, how do you call ...
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0answers
14 views

An element a of a complex normed algebra is Hermitian iff $\|\text{exp}(i\alpha a)\|=1 \, \forall \alpha \in \mathbb{R}$

The (algebraic) numerical range $V(a)$ of an element $a$ of a complex unital normed algebra $A$ is defined as: $V(a)=\{f(a):f \in A^{'}, \|f\|\le1, f(1)=1\}$. $a$ is said to be Hermitian iff ...
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0answers
11 views

Limit of an element in a unital C*-algebra

Let $A$ be a unital C*-algebra. Show that an element $x$ of $A$ is self-adjoint if and only if $\lim_{t\to 0}\frac{1}{t}(||1+itx||-1)$=0. My attempt: Suppose $x=x^*$. By functional calculus of x, ...