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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100 views

How to show that the set of functions $1,x,x^2,x^3…$ is linearly independent?

Show that the set of functions $1,x,x^2,x^3...x^n...$ is linearly independent on any interval $[a,b]$. If $$c_1+xc_2+x^2c_3+x^3c_4...=0$$ we should show $$c_i=0,\quad i=1,2, \ldots$$ how could I ...
1
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0answers
14 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
2
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0answers
31 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
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0answers
22 views

For an inductive limit $X = \bigcup X_n$ of vector spaces, show that $X$ is complete if $X_n$ is complete for all $n$

Let $X$ be a vector space. Suppose that $\{X_n\}_{n=1}^\infty$ is a sequence of vector subspaces such that $X_n \subseteq X_{n+1}$ for all $n$, Each $X_n$ is a locally convex topological vector ...
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0answers
10 views

Topological conditions on compact Hausdorff $X$ under which the unital C* algebra C(X) is separable

If $X$ is a compact Hausdorff topological space, under which extra assumptions do we get that the unital $C^*$ algebra $C(X)$ is separable?
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1answer
36 views

What is the limit of the $n$-th power of the shift operator

Let $T: \ell^2 (\mathbb R) \to \ell^2 (\mathbb R)$ be the left shift operator $(x_1,x_2,x_3, \dots) \mapsto (x_2,x_3,x_4,\dots)$. Let $T^n$ denote a left shift by $n$ positions. What is $\lim_{n \to ...
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0answers
28 views

Proving the existence of limit of an integral

Let $g:\mathbb{R}^d\to\mathbb{R}$ a smooth function and $B:\mathbb{R}^d\to\mathbb{R}^d$ a Lipschitz continuous vector field. I have to study the limit of the following integral ...
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0answers
13 views

Strict convexity and best approximations

Let $V$ be a normed vector space. It is said to be strictly convex if its unit sphere does not contain nontrivial segments. A subset $A \subset V$ is said to have the unicity property if for any $x ...
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0answers
21 views

Compactness of an operator involving the resolvent of laplacian

Let $w\in L^n(\mathbb{R}^n)$, and for $\tau\in\mathbb{C}$, $Im(\tau)\neq 0$, let $R_{\tau}=(-\Delta-\tau)^{-1}$ be the resolvent of the Laplacian. I need to show that $T:=wR_{\tau}w$ is a compact ...
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1answer
18 views

How to finish this proof: invertibility and zero divisorness in $C(\Omega)$

I tried to prove the following and was wondering if someone could please help me finish my proof: Let $\Omega $ be compact Hausdorff. Then $f \in C(\Omega)$ is a left topological zero divisor if and ...
3
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1answer
43 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
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1answer
23 views

$\ell^p$ spaces' inclusion

$$ \ell^s\subsetneq \bigcup_{k<p}\ell^k\subsetneq \ell^p\subsetneq\bigcap_{k>p}\ell^k\subseteq \ell^q $$ for any $1\le s<p<q$. Any idea to prove these inclusions? Counterexamples for the ...
0
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1answer
16 views

Orthonormal basis $L^2(a,a+2\pi)$

Let $$\mathcal{B}=\left \{\frac{1}{\sqrt{2\pi}},\frac{\cos x}{\sqrt{\pi}},\frac{\sin x}{\sqrt{\pi}},\frac{\cos 2x}{\sqrt{\pi}},\frac{\sin 2x}{\sqrt{\pi}},\dots\right \}$$. This is an orthonormal basis ...
-3
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0answers
33 views

Subset Lp space. [on hold]

Let S closed vector space subset $L^1$($\mu$), where $\mu(X) < \infty$. Assume $f \in S \Rightarrow f \in L^p(\mu)$, for some $p>1$. Prove that $\exists p>1$ so that $S \subset L^p(\mu)$?
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2answers
39 views

Matrix of Functions

I was thinking about a problem and I realized for a family of functions say $L^1([a,b])$ one can define matrices with functions as elements: $$ A=\left[ \begin{array}{ccc} f_{11}& ...
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1answer
33 views

Function with compact support

Let $d\geq 1$ and $U \subset \mathbb{R^{d}}$ be open. For any $K\subset U$, compact, there exists $u_{K} \in C_{0}^{\infty}(U)$ with $u_{K} \geq 0$ such that ${\rm supp \,}u_{K}=K $ ? I am looking ...
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1answer
31 views

Explanation of the Gram-Schmidt orthogonalization process

There is a proof of Gram–Schmidt orthogonalization in Kolmogorov's book. Can you explain $h_n$ and how do we write $f_n=b_n\varphi_1+\cdots+b_{n,n-1}\varphi_{n-1}+h_n$? My main question is why does ...
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0answers
32 views

using Stone–Weierstrass theorem for completely regular space

Let X be completely regular. If K is a compact subset of X, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a Locally ...
3
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0answers
47 views

II$_1$-factors with finite commutant and trivial intersection generate $B(H)$?

Let $H$ be an $\infty$-dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Let $\mathcal{A}$, $\mathcal{B} \subset B(H)$ be II$_1$-factors such that $\mathcal{A}'$, ...
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1answer
39 views

What's going on here with zero divisors and invertibility?

I was trying to prove: If $\Omega$ is compact Hausdorff then $f\in C(\Omega)$ is a left topological zero divisor if and only if $f$ is not invertible. Showing $\implies$ is easy. This is an exercise ...
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1answer
33 views

Norms are continuous maps [on hold]

How can I prove that the mapping $$ f:\mathbb{R}^{N}\rightarrow\mathbb{R}, $$ defined by $$ f(\mathbf{x})=\|\Phi\mathbf{x}\|^{2}_{2}, $$ is continuous?
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0answers
24 views

Find the closed-form of a series

Suppose that $x$ is positive number such that $x>0$. I just wonder is there existing a closed form of the series below $f(x)=\sum_{l=0}^{\infty}(2l+1)e^{-xl(l+1)}$. Is the well-known ...
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1answer
31 views

Semantics: Why do we say that a “norm” is induced or derived form inner product when the two are equal?

What do people mean when they say that a norm is induced from an inner product? Why is it not the other way around?
3
votes
1answer
57 views

Show $\lim_{n\to\infty} n^p f(nx) = 0$ exists in the distributional sense

Let $f\in C^\infty(\mathbb R)$ be periodic, with period $2\pi$ and have mean zero ($\int^{2\pi}_0 f(x)dx =0$). Show that for any positive integer $p$ the following limit is valid in the ...
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0answers
32 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
2
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1answer
54 views

Sequence of orthogonal vectors in a Hilbert space

Let $\{x_n\}_{n\in\mathbb{N}}$ be a sequance of pairwise orthogonal vectors in a Hilbert space $H$. Show that the following are equavalent: (a) $\sum_{n=0}^\infty x_n$ converges in the norm topology ...
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0answers
46 views

Prototypical examples of functions in various function spaces

As someone without a firm background in functional analysis, I'm constantly getting confused by the bombardment of different functional spaces that people call upon to explain something (i.e. X true ...
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0answers
30 views

Prove Joint distribution of estimators

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
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0answers
21 views

Basis in Functional Space [on hold]

Show that $\zeta_{r_{0}}=\delta(r-r_{0})$ can be a basis in functional space F. What should I show for the given function to be a basis ?
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1answer
32 views

Question about Closed Graph Theorem

A version of the Closed Graph Theorem states that if $T: X\rightarrow Y$ is a linear operator between Banach spaces then $T$ is bounded iff the graph of $T$ is closed in $X\times Y$. To check if the ...
0
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1answer
34 views

Do we have inequality for this nested $\inf$ expression?

Let $A$ be a unital Banach algebra. I am trying to work out whether $$ \left | \inf_{\|c\|=1} \|ac\| - \inf_{\|d\|=1}\|bd\| \right | \stackrel{\ast}{=} \inf_{\|c\|=1} \inf_{\|d\|=1} \left |  \|ac\| ...
0
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1answer
52 views

How to prove that this inequality holds

Let $A$ be a unital Banach algebra. I wanted to prove the following inequality but didn't manage: $$ \begin{align} \left | \|a\| - \inf_{d \in A: \|d\| = 1}\|bd\| \right | \le \inf_{\|d\|=1} \left ...
2
votes
2answers
71 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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0answers
13 views

L1 expected error for scale/translation in densities

This is an example given in the article about testability (Devroye and Lugosi 1999) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
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1answer
29 views

Inverse of an operator on two functions

I have the following operator, defined for two twice-differentiable functions $f,g$: $X(f,g):=\frac{(g')^3+fg'f''+g'(f')^2-ff'g''}{g'f''-f'g''}$ This operator has the following property: A curve ...
0
votes
1answer
37 views

Measurable function that's defined almost everywhere

If $(X, \Sigma, \mu)$ is a complete measure space, and $f$ is a function that is defined almost everywhere, can I use the language that $f$ is measurable? What does it mean for this function that is ...
2
votes
1answer
24 views

Can we integrate a measurable function defined on a conull subset of a complete measure space?

Suppose $(X, \Sigma, \mu)$ is a complete measure space, and suppose $f$ is a measurable function with domain of $f$ the set $X \setminus N$ for a measurable set $N$ of measure $0$. Does it make sense ...
1
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1answer
22 views

Inductive Limit of directed locally convex Frechet Spaces

Let $\Phi=\bigcup_{i\in \mathbb{N}}\Phi_i$ be the inductive limit of an upwardly directed set of countably-seminormed spaces (i.e. the locally convex topology is given by a countable family of ...
1
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1answer
43 views

A Density Problem

Let $ \mathscr{D}=\mathscr{D}(\mathbb{R}^n - {0}) $ be the space of smooth functions with compact support in $ \mathbb{R}^n - {0} $ topologized by the standard Schwartz topology and let $ \mathscr{C} ...
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0answers
13 views

Quotient space and continuous linear operator

I'm trying to study some arguments of math by myself and I have some problems to understand the interpretation of the norm about linear operators. The books says that there's a correspondence between ...
3
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3answers
77 views

Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
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0answers
24 views

Operator norm with $\inf$

Let $T: V \to W$ be a linear operator. THe operator norm is defined as $$ \|T\| = \sup_{v\in V: \|v\|_V = 1} \|Tv\|_W$$ Does $$ \|T\|' = \inf_{v\in V: \|v\|_V = 1} \|Tv\|_W$$ define a norm? I ...
0
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2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
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0answers
24 views

Applying the Hahn-Banach separation theorem

I have a question applying the Hahn-Banach theorem. I would apply this version of the Hahn-Banach separation theorem. Theorem. Let $V$ be a topological vector space over $\mathbb{R}$. If $A$, $B$ are ...
1
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1answer
22 views

If $K_1,…,K_n$ are compact convex sets then ${\bar conv}(K_1,…,K_n)= conv(K_1,…,K_n)$

If $X$ is a locally convex space and $K_1,...,K_n$ are compact convex subsets of $X$, then ${\bar conv}(K_1,...,K_n)= conv(K_1,...,K_n)$ and this convex hull is compact. Unfortunately I do not have ...
2
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1answer
28 views

Weak convergence and infinite sum

Suppose that $\psi_n$ converges weakly to $\psi$ in a Hilbert space $H$. Assume further $\{\phi_k\}$ is an orthonormal sequence in $H$. Is it plausible that ...
1
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1answer
41 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
2
votes
1answer
54 views

Is a linear operator on $\ell^2$ defined by the inner product necessarily bounded? [duplicate]

If $a=\{a_n\}\in \ell^\infty(\mathbb{R})$ and $\langle a,x \rangle<\infty$ for all $x\in \ell^2(\mathbb{R})$, (where $\langle a, x\rangle=\displaystyle \sum_{k=1}^\infty a_kx_k$), then is $a\in ...
1
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1answer
56 views

Sequence of measurable functions converging a.e. to a measurable function?

I understand if $(X, \Sigma, \mu)$ is a measure space, and we have a sequence of measurable functions $f_{n}$ such that $\lim \limits_{n \to \infty} f_{n}$ exists almost everywhere d$\mu$ (a.e. ...
2
votes
1answer
25 views

Confusion on statement of Fubini's theorem for characteristic function of measurable set

I'm having trouble understanding what this theorem is saying. Theorem. Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ be a complete measure space and suppose $E \in \overline{\Sigma ...