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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47 views

Why $\mathbb R$ is not complete with the metric $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$?

Suppose $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$. Prove that $\mathbb R$ is not complete with this metric. This is exercise 12 from chapter 1 from Rudin's Functional Analysis. ...
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1answer
55 views

Existence of a Frechet topology on the dual of a barreled space

I have a Hausdorff separated locally convex barreled space $(X,\tau)$ with topological dual $X^*$. My questions are: $Q_1$ Is there a topology $\tau^*$ on $X^*$ that is finer than the weak-star ...
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1answer
40 views

A question about orthogonal projection

Here is a quotation of a book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. (P245) Let $H$ be a separable Hilbert space and $\Omega\subset B(H)$ be a separable set and let ...
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0answers
32 views

Gradient of an Inner Product in a more general Vector Space

I was looking at the following question: Differentiating an Inner Product that was talking about the derivative of an inner product to be: $$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), ...
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2answers
34 views

Harmonic functions and Holder's inequality

In the book Real and Complex Analysis by Rudin, it is given that by applying Holder's inequality to the $u(re^{i\theta})=\frac{1}{2\pi}\int_{-\pi}^\pi P_r(\theta-t)f(t)dt$ we get ...
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44 views

If a sequence converges weakly in a closed subspace $M$ of a Banach space, then the (strong) limit point is in $M$.

Let $M$ be a closed subspace of the Banach space $X$ and let $x_{n}\in M$ converge weakly to $x$. Show that $x \in M$. We use the following definition $x_n\rightharpoonup x$ in $X$, $x_n$ ...
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53 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
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2answers
84 views

A question about a conditional expectation in C*-algebra

Let $\Gamma$ be a discrete group. Consider a conditional expectation $\Phi: B(l^{2}(\Gamma))\rightarrow l^{\infty}(\Gamma)$ defined by $$\Phi(T)=\sum_{g\in \Gamma}e_{g,g}Te_{g,g},$$ where $e_{g,g}$ is ...
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1answer
16 views

A question about close line segment in TVS.

Suppose $E$ a topological vector space,which need not be Hausdoff. $x,y\in E$ are different. How to prove the close line segment $\{\alpha x+(1-\alpha)y:\alpha\in[0,1]\}$ is closed. And should it be ...
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1answer
35 views

A question about linear functional on TVS

Let $E$ be topological vector space on field $\mathbb{R}$(or $\mathbb{C}$), which need not be Hausdoff. $f$ is a linear functional on $E$, and there are open set $U\subset E$ and $t\in \mathbb{R}$(or ...
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1answer
35 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
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1answer
55 views

Does a compact operator always have a kernel?

I am sorry if this question is stupid..... I raise it when I read Lax's book Functional Analysis. We know that some integral operators are compact, for example an integral operator from $L^2[Y]$ to ...
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1answer
43 views

The proof of “Every quasidiagonal C*-algebra is stably finite”

Here is a quotation in a book "C*-algebras and finite-Dimensional Approximations" by Nate and Taka (P241). Recall that an isometry $s$ is called proper if $1-ss^{*}\neq0$ Definition 7.1.14 A ...
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27 views

p-direct sum and dual spaces

I have a problem with my assignment of Linear Analysis. It should be rather easy and straight-forward, but I have problems =(. Let E and F be normed spaces. For $p \in [1,\infty]$, define the ...
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1answer
22 views

How to get a smoothing operator from a rapid decreasing function?

From John Roe: Elliptic Operators, topology and asymptotic methods, page 82-83. Let $\mathcal{D}$ be a Dirac operator on the spin bundle $S$, then any section $s\in L^{2}(S)$ has a "Fourier ...
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0answers
15 views

Measurability of the composition of sections and probability measures

Let $X,Y,Z$ be Polish spaces endowed with their respective Borel sigma algebras, and $\Delta(Y),\Delta(Z)$ be sets of Borel probability measures on $Y$ and $Z$, endowed with the weak*-topologies (and ...
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1answer
27 views

Kolmogorov n-width of N+1 dimensional ball

For a normed linear space $\mathscr{X}$, let $\mathscr{A}\subset\mathscr{X}$ and $\mathscr{X}_N$ any $N$-dimensional subspace of $\mathscr{X}$. Define the $n$-width of $ \mathscr{A}$ in $\mathscr{X}$ ...
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27 views

Using function parameters as representation

I was wondering if there is some field of mathematics which analyzes situations where you use function partners as representations, e.g. for classification or regression. For example, let's say I ...
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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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25 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
30 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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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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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 ...
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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
78 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 ...
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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 $k(n) \times k(n)$ ...
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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
53 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
19 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 ...
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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 ...
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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
16 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 ...
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35 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 ...
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122 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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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 $\{s(t)\}_{t\geq 0}$. ...