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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2
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2answers
46 views

If $f,k\in C^{2\pi}$, then $\int_{-\pi}^{\pi}f(x+t)k(t)\mathop{dt}\in C^{2\pi}$

If $f,k\in C^{2\pi}$, then show that $\int_{-\pi}^{\pi}f(x+t)k(t)\mathop{dt}\in C^{2\pi}$ where $C^{2\pi}$ is the space of continuous functions with period of $2\pi$ Thoughts/Attempt: Suppose ...
2
votes
0answers
24 views

$L_{\infty}$ norm of Fejer Integral

How does one show the following? $||\sigma_n(f)||_{\infty} \leq ||f||_{\infty}$, where $\sigma_n(f)(x) = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x+t)K_n(t)\, dt$, and $K_n(t) = ...
0
votes
0answers
24 views

Property of $ L^p( [0,T] , X) $ with X Banach space

I'm starting to study the theory on linear parobolic equation using Evans's book "Partial differential equation". At pag 285, he speaks abaut $ L^p( [0,T] , X) $ with X Banach space. This type of ...
2
votes
1answer
36 views

$l1$ embeds in $X$ Banach space implies $X$ can't be reflexive.

Let $X$ and $Y$ be Banach spaces. We say $X$ is isomorphic to $Y$ if there exists a bijective linear bounded operator $T:X\rightarrow Y$ (Note that by the open mapping theorem the inverse is also ...
-1
votes
1answer
42 views

Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
0
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0answers
50 views

Discrete J-method of interpolation (about understanding theorem statement)

The discrete $J$ method is, given Banach spaces $A_0$ and $A_1$: The interpolationn space $[A_0, A_1]_\theta$ is defined by: $a \in [A_0, A_1]_\theta$ if and only if $a$ can be written as ...
1
vote
1answer
18 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
2
votes
0answers
37 views

When does a linear map become the identify map?

My question is about a linear map defined on the set of smooth periodic functions. Precisely, let $C$ be the set of infinitely many times continuously differentiable and $2\pi$ periodic functions, ...
2
votes
0answers
15 views

Measurability preserving operators on $L^2$

Given $L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$, a $\sigma$-algebra $\mathcal{G} \leq \mathcal{F}$, a function $f \in L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$ which is $\mathcal{G}$-measurable, ...
1
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0answers
30 views

Trace class operator

Let $A\in B(H)$ and $\sum_{E}|\langle A e,e\rangle|< \infty$ for every orthonormal basis $E$. Show that $A$ is a trace class (means $\sum_E \langle |A|e,e\rangle < \infty$). I can not prove it. ...
4
votes
0answers
57 views

why $ \nabla v_n \to \nabla v \ \ (a.e.)$ and $ v_n \to v $

Can someone see the 10th line of page 9 in this article and give a hint that why $$ \nabla v_n \to \nabla v \ \ (a.e.)$$ and $$ v_n \to v $$ and how with theorem 2.1 we could conclude there exists $ ...
8
votes
2answers
139 views

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...
0
votes
1answer
9 views

Is $f(u) := \int_\Omega u^2(x)h(x)$ weakly lower semicontinuous in $L^2(\Omega)$?

Define $f(u) := \int_\Omega u^2(x)h(x)$ weakly lower semicontinuous in $L^2(\Omega)$, where $h \in L^\infty(\Omega)$, but nothing is known about the sign of $h$? I do not believe it is weakly lower ...
0
votes
0answers
37 views

Diagonalizing an operator which consist of eigenvalues of finite type

Let $A$ be a linear unbounded operator. Suppose that the spectra of $A$ consist of finite type eigenvalues only. Thus the eigenvalues are isolated points in $\sigma(A)$ and the generalized spaces are ...
0
votes
0answers
23 views

Proof that pull-back of closed set by continuous functional is weakly closed needs continuity?

Question: In Banach space $X$, if $\phi \in X^*$, then pullback of a closed set is weakly closed? I wrote the following proof: Let $X$ be Banach space. Let $I$ be a closed interval of ...
2
votes
0answers
34 views

quick into 'Function Analysis', ‘Measure Theory’

Can someone suggests some quick introduction document?
2
votes
2answers
54 views

How can I show $U^{\bot \bot}\subseteq \overline{U}$?

Let $H$ be a Hilbert space and $U$ a subspace. Let $U^{\bot}$ denote its orthogonal complement. I had no trouble showing $\overline{U}\subseteq U^{\bot\bot}$. But now I'm stuck for $\supseteq$. ...
0
votes
0answers
8 views

finding number of functions

Find number of functions from set A to set B where cardinality of set A is 'm' and cardinality of set B is n the functions must be having these indicated properties - A) total number of functions B) ...
0
votes
0answers
34 views

Is there a name for functions “opposite in nature” to orthogonal functions?

Suppose a function $f_n(x)$ is orthogonal over some domain $[a,b]$, then we have $$\left|\int_a^b f_n(x)f_m(x)dx\right| \left\{\begin{array}\\>0\text{ if }n=m\\ =0\text{ if }n\neq ...
0
votes
2answers
32 views

$(\mathcal L(X,Y),\|\cdot \|)$ is complete if $Y$ is complete.

In the proof, I get: $$\forall \varepsilon>0,\exists N\in\mathbb N: \forall n\geq N,\forall x\in X, \|(\hat T-T_n)x\|\leq \varepsilon\|x\|$$ where $\hat T$ is such that $T_nx\to \hat Tx$ for all ...
0
votes
1answer
34 views

analysis about elliptic PDEs

I want to study elliptic PDEs,but i have no knowlegde the analysis behind it, such as Arzelà–Ascoli theorem,sobolev embedding,campanato space,Rellich theorem,Poincare inequality... Do you have some ...
1
vote
0answers
27 views

Compact operator and a sot convergent sequence of operators

The following is an exercise of Conway's operator theory: I proved all parts of this exercise except $\|KT_n\| \to 0$. I can easily prove $\|KT_n^*\|\to 0$, but do not have any idea to prove ...
0
votes
1answer
55 views

Norm of bounded real function

Let $X=[0,1]$ and $f$ be a continuous real function defined on $X$. The norm of $f$ is defined by $\Vert f\Vert$=sup$\vert f(x)\vert$ Pls. how do i show that the function $f$ is bounded and for $g$ ...
0
votes
1answer
22 views

Closure of the image is equal to image of $u^\ast u$?

Let $u \in B(H,H')$ where $H,H'$ are Hilbert spaces and let $u^\ast$ denote its adjoint. How can I see that $\overline{u^\ast(H')} = u^\ast u (H)$?
1
vote
1answer
23 views

States: KMS-Condition

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state $\omega$. Does it suffice to have on a dense set the KMS-condition: ...
1
vote
0answers
26 views

strong convexity second-order derivative

When in general that for 2nd order differentiable strong convex function $f(x)$ with modulus $m$, that $x \in \mathbb{E}$ but $\nabla f \in E^*$ (the dual space), does that $\nabla^2 f \succeq m ...
2
votes
1answer
90 views

Show that the operator $(x_n)_n\mapsto (\frac{x_n}{n}) $ is compact

I want to show that the following operator is compact: $$T:\mathbb \ell^p\rightarrow \mathbb \ell^p, \text{ }(x_n)_n\mapsto(\frac{x_n}{n})_n \text{ } 1\leq p<\infty$$ Its the first time that ...
1
vote
1answer
38 views

help to prove $||u||_{W^{2,2}(\Omega) }\le C ||\Delta u ||_{L^2(\Omega )} $

Can some one give a reference or hint for proving $$||u||_{W^{2,2}(\Omega)} \le C ||\Delta u ||_{L^2(\Omega )} $$
0
votes
1answer
12 views

How do I prove that $B(V,W)$ is complete?

Let $V,W$ be banach spaces and $B(V,W)$ be the space of bounded linear operators equipped with the operator norm. How do I prove that $B(V,W)$ is complete? Let $\{T_n\}$ be a Cauchy sequence in ...
0
votes
0answers
16 views

The proof of $(c_0)^* \cong l^1$ always requires construction from $l^1$, not $(c_0)^*$?

Together with the proof that the dual of $l^1$ is $l^\infty$, I understood the element of $l^1$ is the great companion with $l^\infty$, in the sense that $\sum a_nx_n$ absolutely converges, so that: ...
0
votes
1answer
31 views

Question about defintion of inner product space

While practising I came across the following easy question: "Is the space $B(0,1):=\{f:[0,1]\rightarrow\mathbb{R}$ bounded$\}$ an inner product space?" But I'm not quite sure what the correct answer ...
1
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0answers
23 views

Distribution annihilated by a vector field

Let $u$ be a distribution in $\mathcal{D}'(M)$ (the continuous dual of $\mathcal{D}(M) = C_0^\infty(M ; \mathbb{C})$), where $M$ is a smooth manifold. Let also $X$ be a smooth vector field on $M$, ...
0
votes
0answers
14 views

Equality with fourier transform

I have problem with the following equality where the Fourier transform appears: Assume that $u_1,u_2:\mathbb{R}^n\to\mathbb{C}$ are Schwartz functions. Prove that for any $\xi\in\mathbb{R}^n$, ...
1
vote
1answer
24 views

James' theorem—going from the separable case to the general case

Consider the following famous theorem by Robert C. James (1964): Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and ...
1
vote
0answers
17 views

Given a $f : \mathbb{N} \rightarrow \mathbb{R}$ find $D \mathbb{R} \mapsto \mathbb{R}$ so that $f(x)=D(x+1)-D(x)$

Suppose you have a function $f(x)$, $f:\mathbb{N} \rightarrow \mathbb{R}$ now you want to find a function $D_f(x)$, $D_f: \mathbb{R} \rightarrow \mathbb{R}$ so that $f(x) = D_f(x+1)-D_f(x), \forall x ...
1
vote
1answer
25 views

Does this theorem hold for Banach space?

Theorem. Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be an invertible operator. Let $S:H\rightarrow H$ be a bounded operator such that $||S-T||\cdot ||T^{-1}|| < 1$. ...
1
vote
1answer
15 views

Significance of closedness of a subspace when writing a Hilbert space as a direct sum

I read that if $U$ is a closed subspace of a Hilbert space $H$ then we can write $H$ as $H = U \oplus U^\bot$ (the direct sum). What is not clear to me is why $U$ is required to be closed. I thought ...
1
vote
1answer
19 views

Orthogonal complement of the kernel of $u\in B(H, H')$

Let $H,H'$ be Hilbert spaces and $u \in B(H,H')$. Let $u^\ast$ denote the adjoint. I know (and can show) that $(\mathrm{im} u)^\bot = \ker u^\ast$. From this I would deduce that $(\ker u^\ast)^\bot ...
1
vote
1answer
42 views

On Fredholm operator on Hilbert spaces

Let $u: H \to H'$ be a continuous linear operator and $H,H'$ be Hilbert spaces. Let $u^\ast$ denotes its adjoint. By definition, an operator $u$ is called Fredholm if and only if $\ker u$ has finite ...
3
votes
1answer
67 views

Questions about the Fourier transform as a unitary transform

As far as I know, the Fourier transform is a (linear) unitary transform: $T: \textbf{L}^2(-\infty, +\infty) \rightarrow \textbf{L}^2(-\infty, +\infty)$ where the basis functions {$e^{i \omega x} | ...
1
vote
1answer
24 views

Is there a point that lies on the boundary of the unit ball in $\lVert\cdot\rVert_1$, and close to the zero-sequence in $\lVert\cdot\rVert_2$?

I am an engineer who is brushing up some functional analysis. I am curious about the following problem I posed to myself: Consider the sequence space of real-valued sequences that will eventually ...
0
votes
2answers
27 views

Prove $\big|\langle x,y \rangle\big| \space ≤ \space \lambda \cdot \|x\|^2+\frac{1}{4\lambda} \cdot \|y\|^2$ in an inner product space

I want to prove that if I have an inner product space with $\lambda>0,$ then $$\big|\langle x,y \rangle\big| \space ≤ \space \lambda \cdot \|x\|^2+\frac{1}{4\lambda} \cdot \|y\|^2$$ Where should I ...
2
votes
1answer
45 views

parameter operator $A_a$ is compact??

I need some help in this exercise. Let define operator on $ L^2[0,1]$: $$ A_af(x)=\int_{0}^{1}{|x-y|}^{a-1} f(y)dy $$ for f $\in L^2[0,1] $. Prove that A_a is compact for all $a>0$. I see that ...
2
votes
0answers
58 views

Hahn-Banach theorem exercise

Let $X$ be a Banach space (over $\mathbb{R}$) and $u,v\in X$ such that $\|u\|=\|v\|=1$ and $\|2u+v\|=\|u-2v\|=3$. Show that there is $f\in X'$ of unit norm such that $f(u)=f(v)=1$. My idea is ...
2
votes
1answer
42 views

How to prove that a function is irrational?

I need to know how to prove that a given function is irrational. Examples: $$ f(x)=\sqrt{1+x^2} $$ $$ f(x)=\tan(x) $$ Information about the definition of rational and irrational functions would be ...
1
vote
0answers
43 views

Sum of closed subspaces of a Hilbert space is closed

Let $M, N ⊂ H$ ($H$ Hilbert), be two closed linear subspaces. Assume that $\langle u, v\rangle = 0$ $∀u ∈ M$, $∀v ∈ N$. Prove that $M + N$ is closed. Take a sequence $(g_n)\in M+N$ such that $g_n\to ...
1
vote
1answer
18 views

Showing a function to be a norm

I want to prove or disprove that $\parallel (x,y)\parallel=\sqrt{\frac{x^2}{9}+\frac{y^2}{4}}$ is a norm on $\mathbb{R^2}$. Since $\{(x,y):\parallel(x,y)\parallel\leq1\}$ is a convex set, ...
1
vote
1answer
36 views

Orthogonal projection on subspace

Let $\Omega$ be a measure space and let $h : \Omega → [0, +∞)$ be a measurable function. Let$$K = \{u ∈ L^2(\Omega);\ |u(x)| ≤ h(x)\ a.e. on\ \Omega\}.$$ Check that K is a non-empty closed convex ...
0
votes
1answer
26 views

Weyl's law, meaning of the asymptotic formula, does it imply a bound?

Weyl's law states the eigenvalues of the Laplacian behave as $$\lambda_j \sim f(n)j^{\frac 2n}\quad\text{as $j \to \infty$}$$ where $n$ is the dimension. Does this literally mean that, $$\lim_{j \to ...
0
votes
1answer
23 views

Irridicible C*-algebra $A$ implies that projection $p$ is rank one if $pAp=\Bbb C p$

Let $A$ be an irreducible C*-subalgebra of $B(H)$ and $p$ be a nonzero projection in $B(H)$. Suppose $pAp=\Bbb C p$, show that $p$ is rank one. I do not have any idea about it. Please give me a ...