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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Example of Heine-Borel Theorm

For a subset S of Euclidean space R^n S is closed and bounded if and only if S is compact (that is, every open cover of S has a finite subcover). I need an ...
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18 views

Is $\ell_1$ complemented in its double dual $\ell_1^{**}$? (i.e., in $\ell_\infty^*$?)

Quick question, y'all. Is $\ell_1$ complemented in $\ell_1^{**}=\ell_\infty^*$? Yes, I searched Google, and also the standard texts. I can't seem to find an answer, but surely this is known. ...
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14 views

Show that the continuum of elements $e^{i\lambda t}$ forms a complete orthonormal subset of $B^2$.

Let $X$ be the vector space of all finite linear combinations of functions of the form $e^{i\lambda t}$ ($-\infty<t<+\infty$), where the parameter $\lambda$ is real. An inner product in ...
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1answer
24 views

parallelogramm law and inner products

Is the sum of two norms $||.||_1=\sqrt{(.,.)_1}$ and $||.||_2=\sqrt{(.,.)_2}$, where $(.,.)_1$ and $(.,.)_2$ are the quadratic forms of inner products on a normed linear space, again produced by an ...
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10 views

“Asymptotic” $\mathbb{R}$-algebras

Definition. By an asymptotic $\mathbb{R}$-algebra, I mean an $\mathbb{R}$-algebra $F$ of functions $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $$\mathop{\forall}_{f:F}\left[\left(\lim_{x ...
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16 views

What do we call the number that measures how good of an asymptote $g$ is to $f$, and what are the basic results about this number?

Suppose we have a (potentially very complicated) smooth function $f : \mathbb{R} \rightarrow \mathbb{R},$ and we're trying to approximate it (in the limit as the input value goes to $+\infty$) by a ...
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2answers
32 views

“Scalar product” of two Lp spaces

I was reading the book A. Lasotta and M. C. Mackey, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamic", Springer, 1991 On page 27, they defined a ``scalar product'' as follows. Let ...
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13 views

Equivalence of positivity

Let us have complex matrices and their real decompositions as $H=H_1 + \imath H_2$ and $L = L_1 + i L_2$. Further, $H_1\ge 0$ and $H_2$ is skew symmetric. $L = I - P$ where $P$ is some positive ...
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3answers
40 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
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35 views

How to prove the uniqueness of probability measure

Probability essentials P-21 Theorem 4.1 (b) Let $(p_\omega)_{\omega \in \Omega}$ be a family of real numbers indexed by the finite or countable set $\Omega$. Then there exists a unique probability ...
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20 views

Does the “differential” of a unitary representation give continuous operators on the space of smooth vectors?

Let $\pi : G \rightarrow U(H)$ be a strongly continuous unitary representation of a Lie group, $G$, on a Hilbert space, $H$. Let $H_\infty$ be the space of smooth vectors in $H$, those $v$ for which ...
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17 views

Positive-definiteness of a specific function

Is the following function positive-definite $$\varphi(t)=\max\left\{{1-\frac{n-\left|2|t|-n\right|}{2(m+1)},0}\right\}$$ where $0<m<n$ and $0\le |t|<n$. I'm aware of the Bochner's theorem ...
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19 views

Can we study a function from different kernel operation?

Here I am referring kernel as an integral operation.The wikipedia link is this https://en.wikipedia.org/wiki/Integral_transform My question is: consider the function insider the integral $f(t)$ is ...
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26 views

Mathematics- fixed point theory [on hold]

If the operator $T$ on a Banach space $X$ is a contraction mapping, then $T$ has a unique fixed point. The inverse of $T$ has also a fixed point but it is never be a contraction mapping-justify.
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significance and importance of spectral theorem

I have started recently started Operator Theory and have been introduced to the Spectral Mapping Theorem: If $a \in \mathcal{A}$, where $\mathcal{A}$ is a unital Banach Algebra and $f \in ...
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50 views

Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.

Let $C^1([0 , 1])$ be the subspace of $C([0 , 1])$ consisting of the functions that have a continuous derivative throughout $[0 , 1]$. Show that the mapping $\Psi:f → f~'$ from $C^1([0 , 1])$ to $C([0 ...
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31 views

Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other

Let's consider following spaces: $L^2(\mathbb{R}) = L^2(\mathbb{R}, \mathbb{C}, \mu_L)$ --- space of $\mathbb{C}$-valued functions defined on $\mathbb{R}$ for which the square of the absolute value ...
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15 views

Bounded linear functionals in solving PDE

Many theorems in functional analysis, Rietz, Hahn-Banach, for example, are used to find linear functionals in certain spaces. But why are bounded linear functionals useful in solving PDE? This ...
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25 views

How is Baire category theorem used here?

The following is a doubt that arouse from reading this paper by Bandyopadhyay, Jarosz and Rao. Let $E$ be a Banach space and $E^{*}$ be its dual space. Let $e_{0}$ be an element of norm one in $E$ ...
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2answers
51 views

A Banach Space cannot have a denumerable basis:Why is it true?

I came across the following theorem: A Banach Space cannot have a denumerable basis which has been proven in my book. I can't understand why is it true since $\mathbb R$ is a banach space over ...
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1answer
21 views

About closedness and boundedness of $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$

Let $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$. To check which one is true: (a) $H$ is bounded (b) $H$ is closed (c) $H$ is a subspace (d) $H$ has interior points My ...
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25 views

Infinite Hamel basis for Banach spaces

What are some standard examples of Hamel basis for Banach spaces with cardinality >= $\aleph_0$? I tried searching, but couldn't find any.
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37 views

Where can I find this definition of “expected value”?

I need bibliography or some text about this definition: "Define the expected value of a function by: $E_{t}(x(t))=(\frac{1}{t})\int_{0}^{t} x(s)ds$. " I think that it's statistics or functional ...
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1answer
22 views

space of all lipschitz maps is a polish metric space

Suppose that $(X, d_X)$ and $(Y, d_Y )$ are Polish metric spaces. Let $L(X, Y )$ denote the set of all Lipschitz maps from $X$ to $Y$ with the pointwise convergence topology. Show that $L(X, Y )$ is ...
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14 views

Application of Riesz-Markov-Kakutani representation

Let A be finite set, $\{f_i\}_{i=1}^{n}:A\to \mathbb{R}$ non-negative and with the following property for every $\sum \lambda_{i}=1$: for any $g=\sum \lambda_{i}f_i$ there exists $a\in A$ s.t. ...
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2answers
50 views

Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
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30 views

Functional Analysis: Continuity of operators [on hold]

Which of the following operators are continuous: a) $A:L^2 [0,1]\rightarrow L^2 [0,1]$ defined by the formula $\displaystyle (Ax)(t)=\int \limits_{0}^{1} K(t,s) x(s)ds$, where $K(t,s) \in L^2( ...
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33 views

A closed subspace of a separable Hilbert Space is Separable

Suppose $X$ is a Hilbert Space which is separable. Let $Y$ be a closed Subspace of $X$. I need to show that $Y$ is separable. Since $X$ is separable it has a countable dense subset say $M$. Taking ...
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25 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
3
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1answer
21 views

Are all normed linear spaces Hausdorff? What about a bounded subset of a normed linear space?

The proof that the dual space of a normed linear space is complete in proposition 5.4 of chapter on Banach spaces (John Conway, functional analysis) consists of restricting the functionals in the dual ...
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16 views

Banach$^*-$ algebra for two different multiplication

Let $B$ be a Banch $^*-$ algebra and we say $f$ is positive linear functional on $B$ if $f(xx^*)\geq 0$ for all $x\in B.$ Let $B$ be a Banach algebra with two different multiplication operations, ...
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For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property ...
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0answers
20 views

Orthogonal Complement: Families

Problem Given a Hilbert space $\mathcal{H}$. Consider a family: $$A:\Lambda\to\mathcal{P}(\mathcal{H}):\lambda\mapsto A_\lambda$$ Remind that: $$A\subseteq\mathcal{H}:\quad ...
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The convergence of the norm implies the convergence of the sequence via the norm. [duplicate]

I want to prove that: Let $(f_n)$ be a sequence in $L^p$ that converges pointwisely to $f$. Prove that $f_n\rightarrow f$ in $L^p$ iff $||f_n||\rightarrow ||f||$. The "only if" is easy and follows ...
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60 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...
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39 views

Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
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1answer
62 views

this series converges and how can I prove it? [on hold]

I need to prove that this series converges . Note that the series is indexed all integers $ \sum_{k\in\mathbb{Z}}\vert\lambda\vert ^{k}, \; \; \; with \; \vert\lambda\vert<1 $
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1answer
39 views

Definition in Operator Theory

I have started learning some Operator Theory. I encountered the following definition. I would like to know why it is that the $f(z)$ in the integrand and the $f(a)$ are both labelled as $f$ where it ...
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11 views

Kernel, Green function and the functional derivative.

I am pretty new to the subject of differential equations and am reading about Green functions and Kernels for the first time. I am more familiar with functional differentiation and am comfortable with ...
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26 views

Meaning of Finite-dimensional subspace of $C_0(\Omega,\mathbb R^N)$.

I am reading a paper which has something to do with a finite-dimensional subspace of $C_0(\Omega,\mathbb R^N)$. In this paper, it about to compute a value defined as $$ \sum_{n=1}^N \alpha_j\| A_j ...
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19 views

approximate unit of $K(H)$- ordering on $K(H)$ and finite rank operators

Let $H$ be a complex Hilbert space with orthonormal basis $\{e_i:i\in I\}$ . Consider the $C^\ast$-algebra of the compact operators on $H$, $K(H)$. For a finite subset $F\subseteq I$, let $P_F$ be the ...
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1answer
27 views

A question on matrix's eigenvalue problem from Eberhard Zeidler's first volume of Nonlinear Functional Analysis.

I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question 1.5a, he gives as a reference for this question the book by Wilkinson called "The Algebraic Eigenvalue ...
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2answers
26 views

Problem in showing that a norm is a norm on one space, but not on another.

I have the following question from a past paper: "Consider the two sets, $$A=\{g\in C^1([0,1]):g(0)=g(1)=0\}$$ and, $$B=\{g\in C^1([0,1]):g'(0)=g'(1)=0\}$$ both subsets of the vector space ...
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$A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\} \Rightarrow$A is closed [on hold]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
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1answer
21 views

To show that $y$ is the best approximation of $x$ from $G$ i.e $y$ is the unique element of $G$ such that $||x-y||=d(x,G)$

Let $G$ be a closed subspace of a Hilbert Space $H$. For $x \in H$, let $y$ be the orthogonal projection of $x$ on $G$. Then I need to show that $y$ is the best approximation of $x$ from $G$ i.e $y$ ...
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1answer
24 views

Series of positive-definite kernels

Suppose I have a positive definite, shift invariant kernel $k_1(x-y)=k_1(\delta)$. I want to know whether the sum (where $a_n\geq 0$) $$ k(\delta) = \sum_{n=1}^{\infty} a_n k_1(n\delta)\tag{*} $$ is ...
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14 views

What is the mapping of Z-transform?

Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as: $$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real ...
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1answer
34 views

Simple Inequality for Proving Equivalent Besov Seminorms

For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty$, and $h\in\mathbb{R}^{n}$, define the quantity $$I_{p}(h):=\left(\int_{\mathbb{R}^{n}}\left|f(x+h)-f(x)\right|^{p}dx\right)^{1/p}$$ and define ...
4
votes
2answers
61 views

The density of polynomials in the space of continuous functions on the unit ball of $\ell^p$

Let $$B = \{a : \|a\|_p \le 1\} \subset \ell^p(\mathbb{N})$$be the unit ball, endowed with the weak topology. For which $p$, where $1 < p \le \infty$, are the functions of the form$$f(a) = q(a_0, ...
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1answer
41 views

How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1*X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff topological ...