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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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?
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1answer
24 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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15 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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29 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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40 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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23 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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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
28 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 ...
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1answer
29 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\| ...
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1answer
37 views

How to prove that this inequality holds

Let $A$ be a unital Banach algebra. Define $\displaystyle \zeta (a) = \inf_{b \in A: \|b\| = 1}\|ab\|$. I wanted to prove that $|\zeta (a) - \zeta (b) | \le \|a -b\|$. I believe I have a proof but, ...
2
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2answers
66 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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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 ...
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1answer
34 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
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1answer
23 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 ...
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1answer
20 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 ...
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1answer
37 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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12 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 ...
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3answers
72 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
23 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 ...
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2answers
37 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 ...
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1answer
21 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 ...
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1answer
27 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 ...
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1answer
39 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
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1answer
53 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 ...
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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
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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 ...
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0answers
36 views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
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24 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
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1answer
30 views

Bounded from below or not?

if i have that a functional $J$ defined on a Hilbert space is weakly lower semi continuous and coercive is it bounded from below ??? Please help me Thank you
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1answer
11 views

Firmly nonexpansive mapping with the fixed point set same as for given nonexpansive mapping

I found PAMS publication vol. 113, no. 3, 1991 by Ryszard Smarzewski called "On firmly nonexpansive mappings". It is written that "to each nonexpansive T on set C one can associate a firmly ...
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56 views

Show a set is dense in $C(X)$

Let $X$ be a totally discontinuous compact space. Show that the algebra generated by $$\{f_F; ~f_F=\chi_F-\chi_{X/F},F \text{ is a clopen subset of }X\}$$ is dense in $C(X)$. My attempt: Suppose ...
3
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1answer
45 views

Is the set of all Taylor polynomials a vector space?

Let $V$ denote the set of all Taylor polynomials of degree $\leq n$ for a fixed natural number $n$ (including the zero polynomial), regraded as real-valued functions of a real variable. Then is $V$ a ...
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1answer
25 views

Difference isometrically isomorphism and homeomophism.

What is the difference between isometrically isomorphism and homeomorphism?is an isometric mapping is continuous?
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25 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
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1answer
24 views

Show that P is an L-projection iff $P^{*}$ is an M-projection

I have started reading "M-ideals in Banach spaces and Banach algebras", but I stuck on the first page. It says that "there is an obvious duality between L- and M- projections: P is an L-projection ...
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0answers
38 views

Soft Question: Idea behind complete orthogonal system [on hold]

this is my foray into fourier analysis and I'm already daunted by some of the terminology used. I hope some of you can provide me with some answers that are intuitive and approachable. What is a ...
1
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1answer
25 views

Paley projection is not absolutely summing

It is well known that the operator $P: H^{1}(\mathbb{T}) \to \ell_2$, given by $Pf= (\hat{f}(2^{n}))_{n \in \mathbb{N}}$ is bounded and its restriction $P_{|A(\mathbb{D})}: A(\mathbb{D}) \to \ell_2$ ...
6
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1answer
72 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
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1answer
30 views

Trace theorem for Sobolev functions: what is the significance of continuous extension to the boundary?

Why in the proof of the trace theorem in L.C.Evans' book on PDE, he has considered functions in $W^{1,p}(\Omega)\bigcap C(\overline{\Omega})$ when it is enough to choose functions in ...
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1answer
40 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
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2answers
38 views

Introduction to Toeplitz operators

I just finished my undergraduate education in mathematics, and i'm starting a graduate program, and i get interest for learning to work with Toeplitz operators, but i have no background with ...
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1answer
39 views

Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
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0answers
28 views

TT* Duality argument

I am trying to obtain $L^p$ estimate for a system of nonlinear PDE. I do not have $L^2$ to work with for my problem, I heard $TT^*$ argument is useful tool if one wishes to mapp from $L^p$ to $L^p$. ...
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1answer
10 views

Question about Neighborhood basis

In the Simon Reed text, after defining the strong operator topology it is said: "A neighborhood basis at the origin is given by the sets of the form $\{S \ | \ S \in \mathcal{L}(X,Y), ...
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0answers
16 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
0
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1answer
32 views

Exercise about spectrum of selfadjoint operator.

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
1
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1answer
36 views

Operator $Au(t) = \int_0^t e^{t-s} u(s) ds$ (Proof Verification)

Consider the space $C([0,1])$ with $||\cdot||_\infty$ norm. Let $A: C([0,1])\rightarrow C([0,1])$ be the operator defined by $$Au(t) = \int_0^t e^{t-s} u(s) ds.$$ And I am not 100% sure about (c), ...
1
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1answer
30 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...