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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2answers
27 views

Proving functional's continuity

Prove that functional $f:C[a,b]\to \mathbb{R},\ f(x)=\int_a^bx^2(t)dt$ is continuous. Any ideas on how to approach this problem?
4
votes
1answer
28 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
0
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0answers
8 views

Infinite matrix defining a bounded operator on $l^2$

I think I only need some help to clear up the terminology and make sure I understand correctly: I've shown that for a sequence $\{f_{k}\}_{k=1}^{\infty}$ in a Hilbert space $H$, there exists $C>0$ ...
0
votes
2answers
21 views

Proving mapping is contraction

Prove that mapping $B:C[0,\tau]\to C[0,\tau]$. $$(Bx)(t)=\left( \int_0^\tau \sin x(s)ds\right) t, \ t\in [0,\tau], \ \tau >0$$ is contraction mapping if $\tau^2<1$. I want to show that for all ...
0
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0answers
7 views

Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$

Let $a,b$ be elements of a unital C*-algebra $A$ with $0\leq a,b\leq 1$ (e.g., $a,b$ are projections). Is it the case there is a state $\tau$ on $A$ such that $|\tau(ab)|=\|ab\|$? If $ab$ is normal ...
0
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0answers
18 views

Wot topology on $B(H)$ is not metrizable

Let $H$ be a infinite dimensional Hilbert space and $B(H)$ be the space of bounded and linear operators on $H$. I know that weak operator topology (wot) and strong operator topology (sot) are ...
0
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0answers
5 views

Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb R$ so ...
3
votes
1answer
20 views

Lower semicontinuous energy functional on compact space of Lipschitz functions

Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open subset containing $0$ and let $L>0$ be some positive constant. Consider the space $A_{0}=\{f \in C^{\infty}(\overline{\Omega}) \mid f \text{ ...
2
votes
1answer
37 views

Show that this path is differentiable but not rectifiable

My path is defined as follows: $\gamma:[1,1]\rightarrow \mathbb R, \space \gamma(t):= \begin{cases} \ (0,0) & \text{if $t$=0} \\[2ex] t,t^2 \cos (\frac{\pi}{t^2}), & \text{if $t$ $\in$ ...
0
votes
1answer
37 views

What are the hypotheses in Levi's monotone convergence theorem?

Today I read monotone convergence theorem , dominated convergence theorem and fatou's lemma And I need some help We know the dominated convergence theorem in Measure theory In its proof we ...
0
votes
1answer
12 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...
0
votes
0answers
12 views

Unitary G-module

I'm not sure if I understand this sentence correctly: "By a unitary G-module we will mean a Hilbert space W on which G acts by means of a strongly continuous unitary representation". $G$ is a ...
0
votes
1answer
14 views

Convergence in the weak operator topology implies uniform boundedness in the norm topology?

If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology? I ...
2
votes
1answer
14 views

Normal Operators: Construction

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$A:=N(1+N^*N)^{-1}\in\mathcal{B}(\mathcal{H})$$ ...
3
votes
0answers
26 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
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votes
0answers
11 views

show there does not exist a best Approximation element in $E$ [on hold]

Let $c_{0}$ to be the space of sequence which converge to $0$,with $l^{\infty}$ norm,and $$ E=\bigg\{x=(x_{n})\in c_{0}\bigg| \sum_{n=1}^{\infty}\frac{x_{n}}{2^{n}}=0 \bigg\} $$ We know that $E$ is ...
2
votes
1answer
33 views

Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
0
votes
1answer
27 views

Is this sufficient for $f'' \in L^2$?

Let $f \in L^2(0,2\pi)$ be taken such that $f$ and $f'$ are absolutely continuous on $[0,2\pi]$ with $f(0) = f(2\pi)$ and $f'(0)= f'(2\pi).$ Is this sufficient to conclude from this that $f'' \in ...
1
vote
1answer
45 views

A rather abstract strongly continuous semigroup

Define $X$ as the Hilbert space $L^{2}(0,\infty)$ and let the operators $T(t):X\to X$, $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$ I want to show that $(T(t))_{t\ge 0}$ is a ...
127
votes
1answer
4k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
2
votes
0answers
30 views

A question regarding Parseval's identity.

In most books/websites, Proposition 2 (see below) is either stated for a Hilbert space or proved via Riesz-Fischer. Does the follow approach (which seems to work in an inner product space) fall down ...
32
votes
3answers
950 views

Instructive proofs in functional analysis

I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
2
votes
1answer
20 views

For fn(z)= 1/nz, If we make fn(0)= 1, does that make the family of functions bounded?

I have a problem that requires me to use a theorem requiring a bounded family of functions. The family provided that I am supposed to use this theorem for is $f_n (z) = \frac 1 {nz}$ when $z \neq 0$ ...
1
vote
0answers
41 views

Showing that the sequence of functions is not Cauchy

I need to show that $ g_n(x)=x^{1/(2n-1)} $ is not a Cauchy sequence in $C[-1,1] $ w.r.t. supremum norm. I tried to find the maximum of the difference of $g_n$ and $g_m$ by just differentiating but ...
0
votes
1answer
43 views

Prove that a function is decreasing

Let $\left(\,c_m\,\right)_{m \in \mathbb{N}}$ be some coefficients which are all positive natural, $c_0=1$, and they are increasing in $m$. Define $$ f(y) = \frac{\sum\limits_{m=0} c_m \, \, ( y ...
3
votes
1answer
31 views

I need help understanding the proof of Lemma 2.4-1 from Kreyszig's Functional Analysis.

Lemma: Let $\{x_1, \ldots, x_n \}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a number $c > 0$ such that for every choice of scalars $\alpha_1, ...
0
votes
0answers
13 views

C* Algebra, f(x,z)

Let $A$ be a $C^*$ algebra, $x\in A$ and $||x|| < 1$. Let $f(x,z) = (1-x x^*)^{-\frac{1}{2}}(1+zx)$, $|z|=1, z\in \mathbb{C}$, $\mathbb{C}$ is the complex field. How to prove: $$ f(x,z)^* f(x,z) ...
3
votes
2answers
94 views

Lemma 2.4-1 in Erwin Kreyszig's “Introductory Functional Analysis with Applications”: Is there an easier proof?

Here's the statement: Let $\{x_1, \ldots, x_n \}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a real number $c > 0$ such that for every choice ...
4
votes
1answer
131 views

Analogue of Lebesgue differentiation theorem in Orlicz spaces

It is well known that $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all $x\in\mathbb{R}^{n}$. Here ...
1
vote
0answers
19 views

When is it possible to bound a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with $\big|\ f(x_1,x_2,\ldots,x_n)\ \big| \le {\prod}_{i=1}^n h_i(x_i)$

Is there any result that specifies when a multivariate function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ can be bounded (either locally or globally) by a product of some functions $h_i:\mathbb{R} ...
1
vote
0answers
22 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
1
vote
0answers
41 views

How to prove the space $H$ is Banach?

$H$={$f$:$f$ and its derivative are absolutely continuous and squared integrable in $\mathbb{R}$}. The norm is ...
4
votes
1answer
49 views

Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
1
vote
1answer
26 views

Time derivative of logistic function [on hold]

I was wondering whether there is a possible solution to this. If we have function $$ y_t = \frac{x_t}{1+x_t}. $$ given that $x_t>0$ we can represent it as a logistic function $$ y_t = ...
1
vote
0answers
20 views

Weak convergence in $D^{1,p}(\mathbb{R}^n)$ and in $\mathcal{M}(\mathbb{R}^n)$

What it means: $$u_n\rightharpoonup u ~\text{weakly in }~ D^{1,p}(\mathbb{R}^n)\\ |\nabla(u_n-u)|^p\rightharpoonup\mu ~\text{weakly in}~ \mathcal{M}(\mathbb{R}^n)\\|u_n-u|^{p^{*}}\rightharpoonup\nu ...
-1
votes
0answers
23 views

Disjoint simple functions dense in $L^2(I^2, R^2)$

Suppose $$S=\{f\in L^2(I^2,R^2)| \exists h_1(x), f_1(x,y)=h_1(x), a.e. (x,y)\in I^2, \\ \exists h_2(x), f_2(x,y)=h_2(y), a.e. (x,y)\in I^2\}$$ For any $f=(f_1,f_2)\in L^2(I^2, R^2)$ and any ...
3
votes
1answer
32 views

How can I prove that $f$ is inner product function

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have ...
0
votes
1answer
24 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b ...
1
vote
0answers
21 views

$c_{00}$ is a dense subset of $c_0$

I would like to show that $c_{00}$ is a dense subset of $c_0$. I am not sure if I am overly simplifying the argument or even making the right argument for that matter. proof: Suppose that $x \in ...
6
votes
1answer
145 views

An interesting semi-linear PDE problem

Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi linear equation $$ -\Delta u+c(u)=f\;\;\text{in}\;\mathbb{R^n} $$ where $f\in ...
29
votes
2answers
516 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
1
vote
0answers
22 views

What is the Fourier basis for all 2 dimensional functions?

Let us say we have a set of all 2-dimensional functions (E.g. 1 time and 1 space dimension). What is the (Schauder) basis for this set?
0
votes
0answers
15 views

First order elliptic pseudodifferential operator and Sobolev space

Let $P$ be an elliptic pseudodifferential operator of order 1 on a compact manifold $M$. Also let the spectrum of $P$ lie inside $(0, \infty)$. I am trying to see how one could prove that $P : H^1(M) ...
1
vote
2answers
33 views

Swapping series and linear operators

If $T$ is a continuous linear transformation between normed spaces. Under what conditions of $T$ and $(a_n)_n$ we have $T(\sum_{n=0}^\infty a_n)=\sum_{n=0}^\infty T(a_n)$?
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votes
0answers
27 views

Proving a result on a sum of convex functions

Let $P=(p_{1} ,...,p_{n})$ and $Q=(q_{1} ,...,q_{n} )$ be two vectors of probabilities (do not sum to 1 so are not each a distribution). Define a function $F: (0,1)\to(0,1)$ with parameters $P$ and ...
0
votes
1answer
26 views

how can we get Pythagoras from the parallelogram law

When using the definition and properties of the inner product, we get the parallelogram law: $||x+y||^2= \langle x+y, x+y\rangle= \langle x, x\rangle + \langle x, y\rangle +\langle y, x\rangle ...
2
votes
1answer
82 views

Is weak topology first-countable or Lindelöf?

Is the weak topology on a Banach space( or normed linear space ) first countable or Lindelöf? If not, which condition should be added? Definition: A topology space is said to be Lindelöf if its any ...
1
vote
1answer
22 views

Evaluating norm of the operator

I have to calculate norm of the operator $\varphi : l_{1} \rightarrow \mathbb{C}$, where $$ \varphi( (x_n)_{n=1}^{\infty} ) = \sum_{n=1}^{\infty} (-4)^{-n} x_{2n}.$$ My attempt was as follow: Let ...
0
votes
0answers
27 views

Continuous homorphism from $(\mathbf{R},+)$ to group of invertible elements in Banach algebra is differentiable

Let $A$ be a Banach algebra with $1$ and $\varphi\colon \mathbf{R}\to A$ be continuous such that $\varphi(0)=1$ and $\varphi(x+y)=\varphi(x)\varphi(y)$ for each $x,y\in \mathbf{R}$. The claim is ...
0
votes
2answers
35 views

Inequality error possibly. How are two inequalities equal?

Notation: $\underline{x}\in \Bbb R^n,||\cdot||_p =\left(\sum \limits_{i=1}^n |\cdot|^p\right)^{\frac1p}$ $$||\underline{x}||_p\left( \sum \limits_{i=1}^n |x_i + ...