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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Quasi regular Dirichlet form

I'm reading "Introduction to the Theory of (Non-Symmetric) Dirichlet Forms" by Zhi-Ming Ma and Michael Röckner (hereafter, [MR]). In Chapter IV, They give a definition of quasi regular Dirichlet form ...
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11 views

Every point in a Tychonoff Space is contained in a compact set

This should be very elementary, but I just can't see it: Is every point in a Tychonoff Space contained in a compact set? I tried to look for a counterexample and figured that it cannot be locally ...
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1answer
16 views

Prob. 4, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $(e_n)$ be an orthonormal sequence in an inner product space $X$. Then, for every $x \in X$, we have $$ \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 \ \leq \ \Vert x \Vert^2.$$ Now ...
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Prob. 3, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to derive the Schwarz inequality?

Let $\left( e_n \right)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X$, we have $$ \sum_{n=1}^\infty \left\vert \langle x, e_n \rangle \right\vert^2 \ \leq \ ...
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Efficiently compute the bases functions of the Signal

Let there be 4 1D signals such that \begin{cases} x(t)=4\sin(10\pi t) \\ y(t)=8\cos(20\pi t) \\ z(t)=16\sin(30\pi t) \\ m(t)=x(t) +y(t)-z(t) \end{cases} Is there a way to compute ...
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2answers
21 views

Laurent's expansion of a given complex function

A function $f(z)=\frac{\sin{z}}{(z-\frac{\pi}{4})^3}$. Find the laurent's expansion of this function. The annulus is given $0\lt |z-\frac{\pi}{4}|\lt 1$.
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1answer
22 views

$\ker f$ is either dense or closed when $f$ is a linear functional on a normed linear space

Let $f$ be a linear functional on a normed linear space $X$. Prove that $\ker f$ is either dense or closed in $X$. Two possibilities can occur, i.e either $f$ is bounded or unbounded. If it is ...
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1answer
22 views

Integral Operator Contraction

I have the following question: I've found the bound as follows $\lvert T f(x)\rvert \le C\lVert f \rVert(x-a)$ using the fact that K is bounded on the closed square so we have $max \lvert K(x,y) ...
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7 views

Invariance of semigroups

$A$ is the infinitesimal generator of the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ and $V$ is a one dimensional linear subspace of $X$. I want to show that $V$ is $T(t)$-invariant $iff$ $V=span{\{\phi\}}$ ...
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12 views

A curious map on the complex plane

Given a fixed $z\in\mathbb C$, I am considering the map $$ g_z:w\in\mathbb C\mapsto \frac{\bar w-z}{w-z}. $$ Does anybody have seen such a function somewhere ? Any interesting properties ? I feel ...
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11 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 ...
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2answers
34 views

Can $ {L^{1}}(G) $ be a $ C^{*} $-algebra?

Let $ G $ be a locally compact abelian group. Then $ {L^{1}}(G) $ is a commutative algebra when equipped with convolution. Is there an involution $ ^{*} $ on $ {L^{1}}(G) $ so that it becomes a $ ...
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45 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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1answer
18 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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23 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
20 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
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
49 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
45 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
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 ...
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36 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 ...
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1answer
33 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
41 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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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
23 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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17 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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1answer
40 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
21 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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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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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 ...
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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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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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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 ...
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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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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
25 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) = ...
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1answer
22 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 ...
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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
46 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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48 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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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
17 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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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) ...