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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1answer
41 views

If two functionals have the same kernel, then one is a multiple of the other [duplicate]

I would like some help with this exercise. Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$. I expect your suggestions. ...
2
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0answers
26 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
2
votes
1answer
38 views

Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$ ||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}. $$ Let $\{\Omega_n\}$ be a sequence of open ...
2
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0answers
42 views

Taking limit and $\sup$

I was reading a proof for $l^\infty(\mathbb{N})$ is complete. One of the steps is that given $$||x_m - x_n||_\infty = \sup_{i\in \mathbb{N}} | {x_m}_i - {x_n}_i| \leq \epsilon,$$ then for each $I\in ...
0
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1answer
39 views

$A,B\in L(X)$ is positive semidefinition hermitian operators and $A^2=B^2$, then $A=B.$

Please help demonstrate that applies: If $A,B\in L(X)$ is positive semidefinition hermitian operators and $A^2=B^2$, then $A=B.$ where $X$ is Hilbert space (real or complex), and $L(X$) algebra of ...
4
votes
1answer
59 views

Convergence in $C(X)$ is uniform convergence.

I read this the convergence in $C(X)$ is uniform convergence. Where $X$ is compact hausdorff topological space and $$C(X)=\{f:X\to\mathbb{C}\;\mid \; f\ \text{is continuous}\}$$ And ...
1
vote
1answer
32 views

Prove this inclusion: $\bigcup_{k<p}\ell^k\subsetneq\ell^p$

Let $1<p<\infty$. I have to prove that $$ \bigcup_{k<p}\ell^k\subsetneq\ell^p. $$ I am not able to find a counterexample to prove the inequality.
1
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0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
2
votes
0answers
25 views

difference between uniformly convex norms and strictly subadditive norms?

What is the difference between uniformly convex norms and strictly subadditive norms? why we need to define two above concept? how they help us to study Banach spaces? Is the norm induced by ...
3
votes
1answer
21 views

convex weak* sequentially closed subset of a separable Banach space implies weak* closed

I'm studying Conway's a course in Functional Analysis by myself. The following is corollary 6.12.7 of this book. If $X$ is a separable Banach space and $A$ is a convex subset of $X^*$ that is weak* ...
1
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0answers
46 views

Positive part of $y$ with $y\in L^2(0,T; H_0^1(\Omega))$ and $y'\in L^2(0,T; H^{-1}(\Omega))$

Let $\Omega \subset \mathbb R^n$ be a domain, sufficiently smooth. Let $T>0$. Define the space $W(0,T)$ by $$ W(0,T) = \{ y \in L^2(0,T; H^1_0(\Omega)): \ y'\in L^2(0,T;H^{-1}(\Omega)),\ $$ where ...
3
votes
1answer
38 views

Rellich's theorem for Sobolev space on the torus

From John Roe: Elliptic operators, topology and asymptotic methods, page 73: Let $H^{k}$ be the Soblev space defined on the torus $\mathbb{T}^{n}$ with the discrete $k$-norm: $$ \langle f_{1}, ...
0
votes
1answer
63 views

Formulas for Schrödinger unitary groups of operators

Let $\Omega$ an open set of $\mathbb{R}^n$. Consider the Hilbert space $X=L^{2}\left(\Omega\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\Omega)$. Is there any ...
0
votes
2answers
24 views

Space of bounded functions vs. bounded space of functions.

Suppose I have a bounded set of functions, say $B\subset C[0,1]$. What exactly does this mean? I.e. is a bounded set of continuous functions equivalent to a set of continuous bounded functions? For ...
1
vote
1answer
44 views

$L_H(X)$ is real vector space,

Please help demonstrate that applies: $L_H(X)$ is real vector space, where $X$ is Hilbert space (real or complex), and $L_H(X)$ the set of all hermitian operator on $L(X).$ Thanks for your help and ...
1
vote
1answer
38 views

Two linear functionals are equal

Let, $f$ and $g$ be two linear functionals such that ker$f$=ker $g$ and $f(a)$=$g(a)$. Then to prove $f(x)$=$g(x)$.
2
votes
1answer
56 views

Show $T: C([0,1]) \rightarrow C([0,1])$ is compact

Consider $T: C([0,1]) \rightarrow C([0,1])$ defined by $$(Tf)(t) := \int_0^1 \kappa_t(s)f(s)ds,$$ where $\kappa:[0,1]^2 \rightarrow \mathbb{R}$ satisfies the following properties: for all $t\in ...
0
votes
1answer
40 views

How to show $\operatorname{codim}(\operatorname{Ker} f)=1$ if f is linear?

Let L be linear space and $f :L\to \Bbb R(\Bbb C)$ is linear functional. $\DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\codim}{codim}$ $\Ker f$ is a linear subspace and $\codim(\Ker f)=1$ ...
1
vote
0answers
27 views

Elliptic regularity on the torus: reference request

Suppose we work on the two dimensional torus $\mathbb T^2$. Let $L_a^2$ be the space of square integrable functions with zero space average and $H_a^m$ be the corresponding Sobolev space. Suppose we ...
1
vote
0answers
30 views

Different Formulations of Riesz' lemma

Version I: Let $U$ be a closed subspace of the normed space $X$ with $U \ne X$. Also let $0 < \delta < 1$, then there exists $x_{\delta} \in X$ with $||x_{\delta}|| = 1$ and $$ || x_{\delta} ...
2
votes
0answers
40 views

Shorter proof for $T$ compact and $x_n \to x$ weaky then $Tx_n \to Tx$ strongly

I proved that if $X,Y$ are Banach spaces and $T: X \to Y$ is compact and $x_n \to x$ weakly then $Tx_n \to Tx$ strongly. I am now wondering if there is a shorter proof? Here is my proof: Let $x_n ...
0
votes
1answer
31 views

Essential support vs. classical support for a continuous function

The essential support of a function $f:\Bbb R^n\rightarrow \Bbb R$ is defined in the following way: Let's denote $\mathcal A_f=\{\omega \subset \Bbb R^n: \omega \quad \text{open}, \quad f(x)=0\quad ...
1
vote
1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
2
votes
2answers
49 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
2
votes
2answers
124 views

A vector space with countable and uncountable basis at the same time

Let $V$ be a vector space over $\mathbb{C}$. Two self-adjoint, commutable linear operators $\xi$ and $\eta$ act on it. Both of their eigenvectors form a complete set of $V$, but $\xi$'s eigenvalues ...
1
vote
1answer
19 views

Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
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3answers
57 views

Prove that if $T=T^*$ and $\sigma(T)=\{\lambda\}$, then $T=\lambda I$

Show that if $T$ is a self adjoint linear operator on a Hilbert space such that the spectrum contains a single point $\lambda$, then $T=\lambda I$. Then, show this is false if $T$ is not self ...
0
votes
1answer
25 views

Is `Symmetric operator linear`

Today reading a book functional analysis of this sentence stroke. please could you help someone to prove this means i.e. that: symmetric operator is linear, thank you preliminarily. Thanky very much ...
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votes
4answers
80 views

Prove that some topology is not metrizable

Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?
2
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1answer
41 views

Global bounded solution of $u_{tt}=\Delta u-mu+h$ in the Hilbert space $X=H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$

Let $\Omega$ be an open subset of $\mathbb{R^n}$. Consider the linear wave equation $$\begin{cases} \dfrac{\partial^{2}}{\partial t^{2}}u\left(t,x\right)=\Delta ...
1
vote
1answer
26 views

Existence of right inverse.

We know that a surjective continuous linear map $ T : X \to Y$ has a right inverse iff $ \ker(T)$ is complemented. Here $X$ and $Y$ are Banach spaces. Is this result true for locally convex ...
0
votes
2answers
77 views

Which functional analysis book is good?

Which functional analysis book is good ? I am aware of linear algebra, real analysis, measure theory and a little bit of topology. It should be intuitive and with full of motivation.
4
votes
2answers
97 views

Fubini's theorem and $\sigma$-finiteness?

I'm reviewing my analysis notes, and I am really confused about what is meant by $\sigma$-finiteness being a hidden hypothesis of Fubini's theorem. Here is Fubini's theorem as was stated to me: ...
2
votes
0answers
33 views

Adjoint of an operator on $C(X)$

Let $X,Y$ be compact and $\tau: Y\to X$ be continuous. If $A:C(X)\to C(Y)$ such that $(Af)(y)=f(\tau(y))$ is a linear and continuous operator, then show that the adjoint operator $A^*:M(Y)\to M(X)$ is ...
8
votes
2answers
81 views

Ways to calculate the spectrum of an operator

Friends, I am learning some very basic stuff of spectral theory and kind of lost, in some sense. I am trying to find ways to compute the spectra of different operators, when they work and don't work. ...
0
votes
1answer
20 views

How to show this functional is convex

Let M be bounded sequences space. $P(x)=\sup_n|x_n|$ ,$\quad x=(x_1,x_2,x_3...,x_n...)$ $P:M\rightarrow \Bbb R^+$ is convex homogeneous functional. Im not sure but it is homogeneous ...
0
votes
0answers
47 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
2
votes
1answer
51 views

Maximal subspace on which an operator is bounded

Consider the Banach space $X=C[0,1]$ of real continuous function on $[0,1]$ equipped with the supremum norm. Consider the operator $A:D(A)\to X$, $Af=f'$ for each $f\in D(A)=C^1[0,1]$. We can see that ...
1
vote
1answer
45 views

Why is $\|A_n -A\|=\sup |\alpha_i|$ in this proof

I have a question about the paragraph in this book that starts with ''We look next ...'' (bottom half of this page). Why is $\|A-A_n\| = \sup_{j > n}|\alpha_j|$? The operator norm is defined as ...
3
votes
2answers
59 views

Properties of reflexive Banach spaces

I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ...
0
votes
1answer
46 views

Motivation behind the definition of topological vector space

I see in a book the following example as a motivation to define topological vector space. $$+: \Bbb R^n \times \Bbb R^n \to \Bbb R^n$$ defined by $(x,y) \to x+y$ and $$.: \Bbb R^n \to \Bbb R^n$$ ...
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0answers
39 views

basic sequence in the complexification induces a basic sequence in the underlying real space?

This should be easy to prove if it is true, but, alas, what SHOULD be easy is not always easy for me ;) Conjecture 1. Let $X$ be a real Banach space and let $X_\mathbb{C}$ denote its ...
5
votes
1answer
66 views

Show that a subspace of l2 is not complete

I would like to know if this exercise is correct. Let $\Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}$. Show that $(\Bbb R^\infty, \| ...
0
votes
1answer
32 views

Show that an operator is bounded.

Let $\{\alpha_{mn} ;m,n\geq 1\}$ be scalars satisfying a- $M=\sup_n\sum_{m\geq 1}|\alpha_{mn}|<\infty $ , and b- $\sup_n|\alpha_{mn}|<\infty$, then $(Af)(n) = \sum_{m\geq 1}\alpha_{mn} f(m)$ ...
5
votes
1answer
61 views

When is $M+N$ closed

Let $X$ be a Banach space and $M,N$ be closed subspaces. If the range of linear transformation $x\to (x+M)\oplus (x+N)$ from $X$ into $X/M\oplus X/N$ is closed show that $M+N$ is closed. or using ...
6
votes
1answer
70 views

Quotients of topological rings

Let $\varphi\colon R\to S$ be a surjective ring homomorphism and let $R$ be a topological ring. Is there some nice characterization of the finest topology on $S$ for with both $S$ becomes a ...
2
votes
1answer
56 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in ...
0
votes
1answer
22 views

Distribution: $f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$. What is its derivative with respect to the parameter $a$ and the limit as $a\to 0$.

Consider the distribution $$ f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$$ Determine the $a$-derivative of this distribution $$ \left < \frac{\partial f_a}{\partial a},\phi \right> = \lim_{h\to0} ...
0
votes
0answers
72 views

Relationship between $\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$ and $\sum\limits_{n=0}^\infty \frac{a_n^2 x^n}{n!}$

For an analytic function with the property $f^{(n)}(0)=a_n$, we have $f(x)=\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$. This can be extended to $f^{(n)}(x)=\sum\limits_{n=0}^\infty \frac{a_{n+1} ...
1
vote
1answer
19 views

What is the norm in the interface space $L^2(\Gamma)$?

Given bounded open domain $\Omega \subset \mathbb{R}^2=\{x={(x_1,x_2):x_i\in \mathbb{R}}\}$. $\Omega$ is divided by an interface $\Gamma$ into 2 open subdomains $\Omega_1$ and $\Omega_2$ such that ...