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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6 views

Is it already known that every finite normalized equiangular frame is tight?

In the literature, (finite normalized) equiangular tight frames are usually defined as FNTFs (finite normalized tight frames) with an additional "equiangular" condition. But I have noticed that any ...
3
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0answers
25 views

Riesz's 1909 proof of the Riesz Representation Theorem

Frigyes Riesz originally proved the Riesz Representation Theorem on $ C[0,1] $ -- here is his 1909 paper in English (original French). He builds a real valued function $ \text{A} $ on $ [0,1] $ ...
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7 views

Confusion about inclusions of dual spaces

We have the triple inclusions $H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)$, where the second inclusion is not literal but in the sense of distributions. Related to this answer, why is the ...
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0answers
18 views

Composition of function with linear functional is Lipschitz implies function itself is Lipschitz

This is taken from Conway's A course in functional Analysis (p. 98, Exercise 9): If $(S,d)$ is a metric space and $X$ is a normed space, show that if $f:S\rightarrow X$ is a function such that for ...
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1answer
16 views

What is the norm of $H^4(0, 1) \cap H_0^2(0, 1)$?

Let $I = (0, 1)$ and $H_0^2(I) = \{u \in L^2(I) : u', u'' \in L^2(I), u = u' = 0 \;\; \text{on} \;\; \partial I\}$. What is the norm of $$H^4(I) \cap H_0^2(I)?$$ $\|u\|_{H^4(I)} = \Big[ ...
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1answer
23 views

Direct sum of vector space

How can give me two or three example about direct sum of vector space But I want the vector is equipped with an inner product
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1answer
15 views

Selfadjoint Operators: Sesquilinear Form

Given a Hilbert space $\mathcal{H}$. Consider a dense positive form: $$s:\mathcal{D}\times\mathcal{D}\to\mathbb{C}:\quad s(\varphi,\varphi)\geq0\quad(\overline{\mathcal{D}}=\mathcal{H})$$ Construct ...
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4answers
42 views

Are primary ideals always contained in unique maximal ideal?

Just wondering, is this a standard fact? I notice a couple Banach algebra texts define primary ideals in this way. Another question: does this property, i.e. being contained in a unique maximal ideal, ...
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2answers
39 views

For a normed vector space $ E $ and an element $ x \in E $, prove that if $ L(x) = 0 $ for every continuous linear functional $ L $, then $ x = 0 $.

Question. Let $ E $ be a normed vector space. Is it true that for a given $ x \in E $, if $ L(x) = 0 $ for every $ L \in E' $, then $ x = 0_{E} $? One way to prove this is to find an $ L \in E' ...
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0answers
13 views

Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
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2answers
19 views

Some question about orthogonal complement

Let $H$ be a Hilbert space and $Y$ is a closed subspace we denote $Z$ for a orthogonal complement of $Y$ How can I prove that $Z$ is a closed subspace of $X$ (I want to prove subspace and closed ) ...
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12 views

An extension of a functional

Let $M$ be a convex subset of a real normed linear space $X$, and $K_t(M)=\{x: \text{$x$ is an interior point of $M$} \}\neq \emptyset$. $F$ is a subspace of $X$. $K_t(M)\cap F=\emptyset$. Prove that ...
2
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0answers
34 views

$e^{iA}e^{iB}=e^{iB}e^{inA} \Longrightarrow e^{itA}e^{itB}=e^{itB}e^{intA}$?

Let $A$ and $B$ be selfadjoint operators ($A$=$A^\ast$, $B$=$B^\ast$) on a Hilbert space and $n\in\mathbb{N}$ such that $e^{iA}e^{iB}=e^{iB}e^{inA}$, where $i=\sqrt{-1}$. Does then ...
2
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1answer
31 views

How do you formally construct the following proof regarding completeness and vector spaces?

Let $S$ denote the vector space of all finitely nonzero sequences; that is, $X =(X_n) \in S$ if $X_n = 0$ for all but finitely many n. Show that $S$ is not complete under the sup norm $\| X \|_\infty ...
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1answer
36 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
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0answers
22 views

Need help understanding this proof of a certain inequality of $L^p$ norms.

The following theorem and proof is lifted from Folland (Real Analysis: Modern Techniques and their Applications). I am having trouble understanding one single line of the proof: Theorem: Let $K$ be a ...
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1answer
21 views

What is the dual of $H^{-1}(\Omega)$?

The dual of $H^1_0(\Omega)$ is defined to $H^{-1}(\Omega)$. But what is the dual of $H^{-1}(\Omega)$? Is it $H^1_{0}(\Omega)$? I am solving a problem which requires me to use the dual of ...
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0answers
16 views

$A \subseteq B$, $B$ has second category itself and $A$ is convex. Why dose $A$ has second catergory itself?

Let $X$ be topological vector space and $A \subseteq B$. $B$ has second category itself and $A$ is convex. Why dose $A$ has second catergory itself?
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0answers
35 views

If $\ell_1$ embeds into $X$ a separable Banach space, can $X^*$ be separable?

First let's defined embedding: $Y$ embeds into $X$, where $X$ and $Y$ are normed spaces, if there exists a 1-to-1 linear map from $Y$ into $X$ that is bicontinuous. Suppose that $\ell_1$ embeds ...
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1answer
20 views

Let $2H=H+H=H-H$ and $H$ is non empty interior.why $H$ is a neighborhood of 0? [on hold]

Let $X$ be topological vector space and $2H=H+H=H-H$ and $H$ is non empty interior.why $H$ is a neighborhood of 0?
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0answers
13 views

$K$ which is of second category in itself.let $H = K \cap ( - K)$. Why $H$ is non empty interior

Let $X$ be topological vector space.Let $K$ be closed, convex, dense subset of $X$ and $K$ which is of second category in itself. Put $H = K \cap ( - K)$. Why does $H$ is nonempty interior?
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21 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
2
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1answer
36 views

Spectrum of operator $Af(t) = \int_0^{t^2} f(s)ds$ on $L^2[0,1]$

Consider a linear operator $A\colon L^2[0,1]\rightarrow L^2[0,1]$ that acts as follows: $$Af(t) = \int_0^{t^2} f(s)ds$$ The problem is to compute its spectrum. I know that the operator is compact ...
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18 views

operator question

can someone please help me answer this problem: If Y is the eigenspace corresponding to an eigenvalue λ of an operator T, what is the spectrum of T|Y ?justify your answer.thanks
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2answers
28 views

Partial ordering of functions

Let $X$ be the set of all real-valued functions $x$ on the interval $[0,1]$ and let $x \leq y$ mean that $x(t) \leq y(t)$ for all $t \in [0,1]$. Does it define a partial ordering/ total ordering? Does ...
2
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1answer
15 views

Does a vector have to be continuous to fall within a set?

The question asks: explain why $\ f(x) = $ $\ x \over \ x^2 + 4x + 3$ is a vector in $C[0, 3]$ but not a vector in $C[-3, 0]$. I know that $f$ is not continuous on $C[-3, 0]$ at $x = -1$ and $x = 3$. ...
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0answers
17 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
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1answer
22 views

Is there an incomplete normed space which is Asplund?

Can there exist an incomplete normed space which is Asplund?
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2answers
18 views

Are elements of a $C^*$-Algebra strictly positive iff their spectrum is strictly positive?

Let $A$ be a $C^*$-Algebra. An element $a\in A$ is said to be positive iff $a=a^*$ and the spectrum $\sigma(a)$ is nonnegative, ie. $\sigma(a)\subset[0,\infty)$. This is equivalent to $\varphi(a)\ge ...
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1answer
56 views

Invariant subspaces in a Hilbert space

Can someone please help me to answer the following problem? Let $(e_k)$ be a total orthonormal sequence in a separable Hilbert space $H$ and let $T: H \to H$ be defined at $e_k$ by: $T(e_k) = ...
1
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1answer
21 views

Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if ...
2
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0answers
26 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles.

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
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1answer
11 views

Palais–Smale compactness condition

Can someone explain the essence of Palais–Smale compactness condition used in the Mountain Pass Theorem, in particular its weak formulation?
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1answer
14 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
2
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1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
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1answer
44 views

Is the following set of infinite absolutely convex combinations closed?

Let $X$ be an infinite dimensional Banach space and let $(x_n)$ be a weakly-null sequence in X. Let $A:=\{\sum_{n=1}^∞ a_nx_n :(a_n)∈B_{l_1}\}$ , where $B_{l_1}$ is the closed unit ball of the ...
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0answers
47 views

Measure theory theorem [on hold]

So far I couldn't find theorems about equality of measures, I would appreciate book recommendations and help with this theorem. Let A be a family of subsets of Ω stable under intersection. If ...
2
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0answers
36 views

(soft question) why do we study complex measures and complex-valued functionals in modern analysis

Recently I am struggling with "complex" things for my "real" analysis class. We are using Folland's Real analysis, 2nd for text book. It seems that Folland is trying to use complex-valued functions ...
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1answer
16 views

Proving strong stability of semigroup

$X$ is the Hilbert space $L^{2}(0,\infty)$ and let $T(t):X\to X$ with $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$. I want to prove that the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ is strongly ...
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1answer
23 views

Banach algebra norms on $M_n(A)$

Let $A$ be a Banach algebra (not sure whether I need $A$ to be unital). I saw the claim that all Banach algebra norms on $M_n(A)$ with continuous projections on entries are equivalent. How does one ...
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0answers
8 views

Tensor products of Lipschitz functions

I have encountered a problem on which I am sure there is some background, which unfortunately I don't know anything about (so that I don't even know where to start). Let $(M, d_M)$, $(N, d_N)$ be ...
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1answer
12 views

Showing that a bilinear form is coercive

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \underset{I}{\int} u''(x) v''(x) ...
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1answer
24 views

Showing a set is dense in Hilbert space $ L^2 [0, 2\pi]$. [on hold]

Why the set $ \{ f \in C [0, 2\pi]: f(0) = f(2\pi) \} $ is dense in Hilbert Space $ L^2 [0, 2\pi]$?
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1answer
15 views

Invertible operators converging to a noninvertible operator in a finite dimensions: Eigenvalue converge to 0?

I feel like this should be an obvious property, but I want to make sure of it before I use it as the key part of a larger proof: If we have two finite dimensional vector spaces $E,F$ of the same ...
2
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2answers
27 views

A function that integrates to zero against a sequence of weights

Fix any $a\in(0,1)$. Is there a nontrivial continuous function $f:[a,1]\to\mathbb R$ so that $$ \int_a^1t^{-2n}f(t)dt=0 $$ for all integers $n\geq0$ and $f(a)=f(1)=0$? I would prefer explicit ...
1
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1answer
27 views

Criterion for Isometry

Let $X$ be a topological vector space, with $d$ an invariant metric compatible with the metric. Let $f:X\to X$ be an involutive linear isomorphism. How do you show that $f$ is an isometry? I ...
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0answers
38 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...
3
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3answers
190 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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0answers
21 views

Problem on The Multiplication Operator

Let $f$ be a bounded measurable function on $X$ and $M_f$ be the multiplication operator on $L^2$($\mu$).Then prove that $\int$ $fdP=M_f$ where $P$ is the $spectral$ $measure$. I have been trying for ...
2
votes
1answer
29 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...