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
160 views

Inequality in a Hilbert space

This was a homework question a couple weeks ago that I couldn't solve. I'd appreciate a solution. Let $\mathcal{H}$ be a Hilbert space and $\{\eta_n \}_{n \in \mathbb{Z}}$ a set of not ...
2
votes
1answer
50 views

Inequality involving a sequence in Hilbert space

Let $H = L^2(0,T;V)$ where $V$ is separable in $H$. We have $y_n(t) = \sum_{i=1}^n c_{i,n}(t)b_i$ where $b_i$ are basis vectors in $V$. Suppose we have the estimate $$\lVert y_n \rVert_H^2 = \int_0^T ...
2
votes
3answers
446 views

Boundedness and pointwise convergence imply weak convergence in $\ell^p$

Let $p\in(1,+\infty)$ and consider the space $\ell^p$ with its usual norm. The following are equivalent: (1) $x_n \rightharpoonup x$ (i.e. $x_n$ weakly converges to $x$); (2) $$\exists ...
2
votes
1answer
270 views

Quotient space of $l^p$ that isometrically isomorphic to $l^p$

Let $Y=\{x\in l^p:x(2n)=0\}$, $1\leq p \leq \infty$. It can be proved that $Y$ is closed subspace of $l^p$. Define the quotient space $l^p/Y=\{x+Y:x\in l^p\}$. Then, by the fact that $Y$ closed, a ...
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1answer
168 views

Surjective map from quotient space of Banach Space that satisfy certain condition

Let $X$ be a Banach Space and $Y$ be a closed subspace of $X$. Define the quotient space of $X$ as collection $\left\{ \left\{ x+y:y\in Y\right\} :x\in X\right\} $. Can we find a a surjective linear ...
1
vote
1answer
110 views

How to calculate norm of operator in Hilbert space

LEt $H_1,H_2$ are two Hilbert spaces. $\{e_1,\ldots,e_n\}\subseteq H_1$ and $\{f_1,\ldots,f_n\}\subseteq H_2$ two orthonormal systems. $\lambda_1,\ldots,\lambda_n\in\mathbb K$. Let $$ U:H_1\to ...
10
votes
1answer
513 views

Why no trace operator in $L^2$?

We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe. I'm sure either ...
2
votes
1answer
248 views

Kernel of the differential operator

I've read somewhere that the kernel of a linear map is closed iff the map is bounded. Consider the derivative operator $D: \mathcal C^1([0,1],\mathbb C) \to \mathcal C([0,1],\mathbb C)$, i.e. $Df = ...
1
vote
2answers
80 views

Basis, dense subset and an inequality

Let $V \subset H$, where $V$ is separable in the Hilbert space $H$. So there is a basis $w_i$ in $V$ such that, for each $m$, $w_1, ..., w_m$ are linearly independent and the finite linear ...
4
votes
1answer
137 views

Prove this inequality from functional analysis

I want to prove this equality used in out lecture notes: Let $D=(0,r)^2 \subset \mathbb{R}^2, r\geqslant 0$. Then, for any $u \in H^1(D)$, there holds $$\lVert u\rVert \leqslant \frac 1 r ...
0
votes
1answer
46 views

Dimension of the range of a differentiable map

For a problem I'm working on I have two Banach spaces $X, Y$ and an injective immersion $T\colon X \to Y$ (that is, a $C^1$ injective mapping having the property that its (Fréchet) differential $dT ...
0
votes
1answer
158 views

Hilbert space the trace

I need help from someone to solve this problem. Given a bounded sequence $(\lambda_n)$ in $\mathbb С$ define an operator $S$ in $B(\ell_2)$ by $S(x_1) = 0$ and $S(x_n) = \lambda_n x_{n-1}$ , ...
2
votes
1answer
143 views

Operators in Hilbert space ( orthogonal projection)

Some one help me to solve this problem i put some hints Show that every projection $P \in \mathcal{B}(H)$ is and extreme point in the convex set $$B_+ = \{T \in \mathcal{B}(H) : T \geq 0, \Vert ...
2
votes
2answers
197 views

Lipschitz property and Lipschitz extension

Is there a Lipschitz function $f$ from a subset of a metric space $U$ to a complete metric space $V$ that has no Lipschitz extension to the whole space $U$?
1
vote
1answer
44 views

Continuity of a restriction operator

Let $\Omega$ be a domain of $R^n, \; n\ge 1.$ I am wondering about the existence of an open subset $\omega$ of $\Omega$ such that $$ \|y\|_{L^\infty(\omega)} \le C \|y\|_{H^2(\Omega)},\; \forall ...
4
votes
1answer
1k views

Basic Open Problems in Functional Analysis

Hello I was wondering if there exists open problems in functional analysis that don't require too much machinary for studying them, I mean, problems that don't require high level prerequisites.. Does ...
6
votes
4answers
399 views

Survey papers for PDE?

I want to know if there is a good website which allows you to download survey papers on PDEs? The "survey" should include a summary of methods, skills, developments etc. I wish to get some basic (or ...
1
vote
1answer
119 views

Uniform contraction proof

Prove that for every uniform contraction function $f$ there exists a unique real $z$ such that $f(z)=z$. A function $f:\mathbb R\to\mathbb R$ is called a uniform contraction if there exists an $a$ in ...
3
votes
0answers
59 views

An example of a space $X$ such that every L-subset of $X^*$ is weakly precompact but not relatively weakly compact

A bounded subset $A$ of $X^*$ is called an L-set if each weakly null sequence $(x_n)$ in $X$ tends to zero uniformly on $A$. The space $X$ has the Reciprocal Dunford-Pettis property if every L-subset ...
1
vote
1answer
490 views

Right Shift Operator

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T:H\rightarrow H$ such that $Te_n=e_{n+1}$, for $n=1, 2, ...
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vote
2answers
130 views

Is the inclusion map of $\ell^1(N)$ in $\ell^2(N)$ bounded and dense?

I am looking for an idea to prove if the inclusion map from $\ell^1(N)$ to $\ell^2(N)$ bounded and does it have a dense image. And why is the set $A:=\{x: ||x||_1\le 1\} \subset \ell^2(N)$ closed and ...
0
votes
1answer
102 views

Absolute continuity and integration formula (explain a statement please)

I read this: For $v$, $w$ in $L^2(0,T;H^1(S))$ (with weak derivatives in $H^{-1}(S)$ for each time), the product $(v(t), w(t))_{L^2(S)}$ is absolutely continuous wrt. $t \in [0,T]$ and ...
2
votes
2answers
131 views

Weak Convergence of Positive Part

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain and $p\in (1,\infty)$. Suppose $u_n\in L^p(\Omega)$ is such that $u_n\rightharpoonup u$ in $L^p(\Omega)$. Define the positive part of $u$ by ...
2
votes
1answer
287 views

Showing boundedness and a coercivity condition for a bilinear form

Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to ...
2
votes
1answer
114 views

Fixed point of a non linear contraction in a convex set

Hi I'm stuck on the following problem of Haim's functional analysis book. Let $C\subseteq H$ ($H$ a Hilbert space) be a non-empty closed convex subset and let $T:C\rightarrow C$ be a non linear ...
0
votes
2answers
158 views

Needing an example of one riemann integrable function

This is easy, but I couldn't find some example of a function that is not integrable but its Riemann improper integral exists and is finite
1
vote
1answer
375 views

About the measurable subsets and the Lipschitz condition

I have, again, a doubt with the measurable subsets. If I have that $T\colon\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is Lipschitz, does $T$ send Lebesgue measurable sets in Lebesgue measurable sets. ...
1
vote
1answer
70 views

About the properties of Lebesgue measurable subsets

This is a doubt about Lebesgue measurable subsets If I have two Lebesgue measurable subsets $E_1, E_2$ in $\mathbb{R}$, is the subset $E_1\times E_2$ Lebesgue measurable in $\mathbb{R}^2$?, If it ...
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vote
0answers
103 views

About Lebesgue measure

This is a problem of Lebesgue measure and measure theory specifically. Suppose that $f:\mathbb{R}^2\longrightarrow [0,\infty)$ is measurable. $\Omega_1\subseteq \mathbb{R}^2$ is Lebesgue ...
0
votes
1answer
95 views

How to prove or disprove this proposition: convex combination of two convex function is non-negative (under some assumption)

Let $f,g$ be two convex function on $D\in\mathbb{R}$ (what about $\mathbb{R}^n$?) satisfying that there is no point $x$ in $D$ such that $f(x)<0$ and $g(x)<0$ at the same time. I want to prove ...
3
votes
1answer
135 views

Question about weak derivative, need explanation of this paragraph

Given the Gelfand triple $V \subset H \subset V^*$, we for for $y \in L^2(0,T;V)$ also $y \in L^2(0,T;V^*)$ and thus $y \in L^1(0,T;V^*)$. I understand this because if $y(t) \in V$ then $y(t) \in ...
0
votes
2answers
85 views

Non-barreled topology compatible with the duality

Given $(X,s)$ a (real) barreled locally convex space (that is, every closed convex and absorbing set in $(X,s)$ is a neighborhood of the origin), is there a (strictly) finer, non-barreled linear ...
2
votes
1answer
141 views

Closed subspace of $L^1[0,1]$

The statement I need to prove is following. Let $S$ be a closed subspace of Lebesgue space $L^1[0,1].$ Assume that for every $f\in S$ there exists a number $p(f)>1$ such that $f\in L^{p(f)}[0,1].$ ...
2
votes
2answers
463 views

Spectrum and point spectrum of this operator

Let $T\in \text{Aut}(\ell^2(\mathbb{C}))$ and $T(x)=(a_1 x_1, a_2 x_2,\ldots)$ where $a=(a_i)_i \in \ell^\infty(\mathbb{C})$. How can I easily see what is $\sigma(T)$ and $\sigma_p(T)$ (that are ...
0
votes
1answer
274 views

Banach space and bounded below operators

Let $T\in L(X,Y)$, where $X, Y$ are Banach spaces. How can I conclude from $\ker(T) = \{0\}$ and $\mathrm{im}(T)$ closed that $T$ is bounded below, e.g. there exists a $c>0$ such that ...
1
vote
2answers
234 views

$Conv(Ex((C(X))_1))$ is dense in $(C(X))_1$?

Let $X$ be compact Hausdorff, and $C(X)$ the space of continuous functions over $X$. Denote the closed unit ball in $C(X)$ by $(C(X))_1$, then it can be shown $f$ is an extreme point of $(C(X))_1$ if ...
3
votes
1answer
52 views

Continuous Connection?

Consider two compact convex sets $C_1, C_2 \subset \mathbb{R}^n$ such that $C_2 \subset C_1$. Let us denote by $\partial C_1$ and $\partial C_2$ their boundaries, that satisfy and $\partial C_1 \cap ...
1
vote
2answers
241 views

operator norm of this multiplier operator

I am having some trouble with some basic properties of a given operator. Firstly, the operator T is defined as taking the fourier inverse transform of the function ...
5
votes
1answer
1k views

An operator has closed range if and only if the image of some closed subspace of finite codimension is closed.

Let $B$ be a Banach space, $H,K$ be closed subspaces and let $K$ be finite dimensional. Suppose $B = H\oplus K$ and $T:B\to B$ is a bounded linear operator. How do I show that $T(B)$ is closed ...
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0answers
73 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
4
votes
1answer
121 views

$(X \oplus_p Y)^*$ isometric to $(X^*\oplus_q Y^*)$

Let $X,Y$ be Banach spaces. For $1 < p < \infty$, define a norm on $X \oplus Y$ by $\|(x,y)\|_p=(\|x\|_X^p+\|y\|_Y^p)^{1/p}$. Homework asks to prove that: $(X \oplus_p Y)^*$ is isometric to ...
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vote
2answers
94 views

Operator T with rank T=1

Let $h,g$ in Hilbert space $H$. Define $T:H\rightarrow H$ by $Tf=\langle f,h\rangle g$. Would you help me to prove that $\dim(\operatorname{ran}(T))=1$. Next, show that If $T$ is finite rank, then ...
2
votes
0answers
55 views

Continuous linear functional on $\mathbb{C}^A$ with the product topology

My homework: Let $A$ be a set. Endow $\mathbb{C}^A$ with the product topology. Then any continuous linear functional $\Lambda:\mathbb{C}^A\rightarrow\mathbb{C}$ is of the form $\Lambda(f)=\sum_{i=1}^n ...
2
votes
2answers
802 views

No Nonzero multiplication operator is compact [duplicate]

Let $f,g \in L^2[0,1]$, multiplication operator $M_g:L^2[0,1] \rightarrow L^2[0,1]$ is defined by $M_g(f(x))=g(x)f(x)$. Would you help me to prove that no nonzero multiplication operator on $L^2[0,1]$ ...
0
votes
1answer
152 views

The notation $ C^0 ([0,T], H^{s} (\Bbb R^n ) ) $

Does the notation $ u \in C^0 ([0,T], H^{s} (\Bbb R^n ) ) $ imply $$ \lim_{k \to \infty}\| u(t_k ) - u(t_0 ) \|_{H^{s} (\Bbb R^n )} = 0 \;\;\text{if} \;\; \lim_{k \to \infty} t_k = t_0$$ as well as ...
3
votes
1answer
497 views

Convergence in distribution (weak convergence)

Let $X_n$ and $X$ be random variables taking values in the metric space $(S,d)$. The sequence $(X_n)_n$ is convergent to $X$ in distribution (or weakly) if $E[f(X_n)] \to E[f(X)]$ for all $f:S\to R$ ...
1
vote
0answers
391 views

Dirichlet and Neumann boundary conditions Finite Element Method

I have the following problem in Finite Element Method $$ -(\alpha u')' + \beta u' + \gamma u$$ with $ \Omega = (0, 1)$, $ u(0) = 0 $ and $ u'(1) = 3 $ to be able to write the weak formulation of ...
1
vote
1answer
777 views

Weakly sequentially compact sets

From Peter Lax Functional analysis page 104: Show that a weakly sequentially compact set is bounded. Definition. A subset $C$ of a Banach space $X$ is called weakly sequentially compact if ...
5
votes
1answer
267 views

solution of Lagrange differential equation are square integrable

I was recently posing myself this question. Given the Lagrange DE $$[(1-x^2)u']'+\lambda u=0,$$ where $\lambda$ is a real parameter and $x\in[-1,1]$, it is well known that, if $\lambda=n(n+1)$ for ...
2
votes
0answers
677 views

Distributions supported on a single point

Let $d=1$. (i) Show that if $\lambda$ is a distribution and $n\geq1$ is an integer, then $\lambda x^n=0$ if and only if $\lambda$ is a linear combination of $\delta:=\delta_{\{0\}}$ and its first ...