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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39 views

$f(x)f(1/x)=f(x)+f(1/x)$

Find a function $f(x)$ such that: $$f(x)f(1/x)=f(x)+f(1/x)$$ with $f(4)=65$. I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$ which leaves $f(1/x)$ as: ...
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1answer
4 views

Sesquilinear Forms vs. Selfadjoint Operators

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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3answers
18 views

Are primary ideals always contained in unique maximal ideals?

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
35 views

For a normed vector space, prove, without the Hahn-Banach Theorem, that if $ L(x) = 0 $ for every continuous linear functional $ L $, then $ x = 0 $.

Let $E$ be a normed space. Is it true that if $L(x_0) = 0$ for every $L \in E'$ then $x_0=0$? One way to prove it is to consider $L \in E'$ s.t. $\|L\|= 1 $ and $L(x_0)= \|x_0\|$ (Existence of such ...
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0answers
10 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
14 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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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 ...
2
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0answers
33 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
votes
1answer
30 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
19 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
33 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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0answers
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
32 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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0answers
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 ...
1
vote
1answer
21 views

Is there an incomplete normed space which is Asplund?

Can there exist an incomplete normed space which is Asplund?
2
votes
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 ...
1
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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
vote
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
25 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?
0
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1answer
13 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
votes
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) ...
1
vote
1answer
39 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
votes
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 ...
1
vote
1answer
15 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 ...
0
votes
1answer
22 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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votes
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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votes
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
votes
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 ...
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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
37 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
votes
3answers
188 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
28 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 ...
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votes
2answers
75 views

Prove $\ell^5$ is contained in $\ell^6$.

I am struggling with the proof to show that, for any $p$, $r$ such that $1 \le p <r < \infty$, that $\ell^ p\subset\ell ^r$. Could somebody please give a helpful nudge by showing how this ...
-2
votes
1answer
31 views

Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
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votes
1answer
36 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
0
votes
1answer
24 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
1
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0answers
16 views

Calculating the norm of a case dependent function

If $X$ is the Hilbert space $L^2(0,\infty)$ equipped with the inner product $$\langle f,g\rangle :=\int_0^\infty f(\zeta)\overline{g(\zeta)}(e^{-\zeta}+1) \, d\zeta,$$ and the operator $T(t):X\to ...