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
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
39 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
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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 ...
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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
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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
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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
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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
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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
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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
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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
27 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 ...
1
vote
0answers
29 views

Bornological/Barrelled Operator-Topologies?

I'm looking for results concerning the following questions. If those have been already addressed in the literature, it would be nice to know proper citations: Let $(E, \tau_E)$ and $(F, \tau_F)$ be ...
3
votes
2answers
39 views

Hilbert space Inequality

Let $K ⊂H$ be a nonempty closed convex set, $H$ Hilbert. Let $f ∈H$ and let $u=P_Kf$. Prove that $$||v − u||^2 ≤ ||v − f ||^2 − ||u − f ||^2, ∀v ∈ K$$ I've tried to use the parallelogram identity and ...
1
vote
0answers
28 views

Weakly closed $\iff$ closed using the Separation theorem

My question is about the following problem. $X$: Banach space, $C$: convex subsets of $X$. Then, followings are equivalent. i) $C$ is closed. ii) $C$ is weakly closed. I ...
1
vote
1answer
30 views

Limit of an average integral?

Lebesgue's differentiation theorem states that if $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is locally summable then \begin{align*} \frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}f(x)\,dx\rightarrow f(x_0), ...
1
vote
2answers
47 views

How to define orthogonal complement in an arbitrary vector space

In this article about codimension there is the following remark: The codimension of a subspace $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all ...
1
vote
1answer
21 views

Orthogonal complement of a subspace in a Hilbert space

In this question it is stated that if $W$ is a closed subspace of $V$, we may define $W^\perp$ to be the subspace of all vectors $v \in V$ such that $\langle v | w\rangle =0$ for all $w \in W$. ...
2
votes
2answers
58 views

Prove that the set $X+Y$ is closed

Consider two sets: $$X=\bigl\{(x,y)\in \mathbb R^{2}:xy=1\bigr\}$$ $$Y=\bigl\{(x,y)\in \mathbb R^{2}:|x|\le 1,|y|\le 1\bigr\}.$$ We find that , both the sets are closed. But, is the set $X+Y$ ...
1
vote
3answers
36 views

Size of function spaces

For example, how big is the space $ C^k(\mathbb{R},\mathbb{R}) $ ? How much is, say, $ C^0 $ larger than $ C^1$ ? How can one figure out ?
0
votes
0answers
50 views

Laplace Transform of $e^{a t^2}$

What is the Laplace transform of $e^{a t^2}$, for positive $a$? In order for Laplace transform to exist function must be locally integrable. Since integral of any compact set $e^{a t^2}$ is finite ...
1
vote
1answer
44 views

Is Fourier transform method suitable for solving equation $\int g(x-t)e^{-t^2} dt = e^{-a|x|}$

Is Fourier transform method suitable for to solve the following equation \begin{align*} \int g(x-t)e^{-t^2/2} dt = e^{-a|x|} \end{align*} Suppose we take the Fourier transform of the above ...
0
votes
0answers
28 views

Why $\mathcal{D}(\Omega)$ be a topological vector space is important?

Let $\Omega\subset\mathbb{R}^N$ be an open set and $\mathcal{D}(\Omega)$ the set of all infinitely differentiable functions with compact support on $\Omega$. In the study of PDEs, we use the ...
3
votes
0answers
27 views

Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
2
votes
1answer
25 views

Compact operators and essential spectral radius

Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra ...
1
vote
2answers
26 views

Proving compactness of an operator

I'm having a hard time finding a solution for the following problem: Prove that the operator $ T \in \mathcal{L}(\ell_2) $ defined with the formula $$ T((x_1, x_2, \dots, )) = (0, x_1, x_2/2, x_3/3, ...
0
votes
1answer
31 views

Complement of the Image is the Image of the Complement

Given a continuous linear map $T:E\to F$ where $E,F$ are normed vector spaces, I am wondering about the trivial question whether for any subset $U\subset E$, it holds or not: $$T(U^c)=T(U)^c$$ I could ...
-1
votes
1answer
21 views

Strict Bessel inequality in $\mathcal{l}^2$

I'm asked to give an example of an $x\in \mathcal{l}^2$ s.t. $$\sum_{j=1}^{\infty}|<e_j,x>|^2<||x||^2$$ Where $(e_j)$ is some orthonormal sequence. However, I think this question is more ...
0
votes
1answer
27 views

General Fourier coefficients and smoothness

Suppose $f\in L^2([0,1],\lambda)$. Are there assumptions on the smoothness of $f$ which translate into the particular behavior of Frourier coefficients. Namely, I have arbitrary complete orthonormal ...
1
vote
1answer
42 views

Is the dual space of all Radon measures the space of signed measures on a $\delta$-ring?

Consider the Banach space $C_c(\mathbb{R})$ of continuous functions with compact support equipped with the uniform norm $||f||_\infty := \sup_{x \in \mathbb{R}} |f(x)|$. Then it is known (Riesz ...
3
votes
1answer
130 views

On a proof of Riesz-Fischer Theorem

Questions : [See below for context.] $\rm\color{#c00}{a)}$ First, is the proof presented below $100$ % correct ? $\rm\color{#c00}{b)}$ How would one justify the LHS of $(2)$ ? Are my ...
0
votes
0answers
89 views

Homotopy classes of $*$-morphisms and unital $*$-morphisms

Let $A$ and $B$ be C*-algebras (non necessarily unital). A homotopy between two $*$-morphisms $\phi,\psi:A \to B$ is a $*$-morphism $A \to C([0,1],B)$ such that you can recover $\phi$ and $\psi$ from ...
1
vote
1answer
36 views

is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$?

Consider the $L^p$ spaces. is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$? is it true if the domain of $L^p$ is finite measure? Thanks
1
vote
0answers
15 views

Coercivity of a bilinear form

Consider a very smooth open and bounded set $\Omega$ of $\mathbb{R}^d$. One can prove that for all $\epsilon >0$, $\exists K_{\epsilon}>0$, such that for all $u \in H^1(\Omega)$ : ...
1
vote
2answers
47 views

Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
3
votes
1answer
29 views

Is $x\mapsto \| Tx\|$ lower semi-continuous?

Suppose $T:\mathcal D(T)\rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$. Is it true that $$ \|Tx\|\leq \liminf_{n\rightarrow\infty} \|T ...
1
vote
2answers
27 views

Norm of Functional on : $c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$

Take $E = c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$ and define on $E$ the functional: $$F(x) = \sum_{n=1}^\infty \frac{x_n}{2^n}$$ $\cdot$Show that $F$ is a linear continuos functional on ...
2
votes
0answers
46 views

Lagrange Multipliers for linear functionals

Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem $$max~f(x)\quad s.t.~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can show ...