2
votes
2answers
32 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
2
votes
2answers
46 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
1
vote
1answer
27 views

Equivalent Definitions of the $L_2$ inner product.

If $g \in L_2(\mathbb{R})$, then we can define the $L_2$ norm to have the following relationship: $\|g\|_2^2 = \int_\mathbb{R} g^2$. If $A\subseteq \mathbb{R}$, then we can define the norm of $L_2(A)$ ...
0
votes
0answers
29 views

give an example of $f \in H^1 (\mathbb R^2)$ [duplicate]

can some help me how to give an example of $f \in H^1 (\mathbb R^2)$ such that $\|f\|_\infty = \infty$. $f\in H^1(\mathbb R^2)$ if $f\in L^2(\mathbb R^2)$ and $f_{x_1},f_{x_2}\in L^2(\mathbb R^2)$ ...
2
votes
1answer
77 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
0
votes
1answer
33 views

how to show this function doesn't belong to Hilbert space?

I am trying to show $\chi_{B_R(0)}(x) \notin H^1 (\mathbb{R}^n)$ , ∀R>0. since $H^1 (\mathbb{R}^n) := W^{1,2}(\mathbb{R}^n)$ That is, I have to show that $\chi_{B_R(0)} (x) \notin ...
-1
votes
1answer
76 views

Hilbert space $L^{2}(0,\pi)$

I wanted to know how I should proceed if I wanted to prove that the closed subspace of $L^{2}(0,\pi)$ generated by {$\sin(kx): k=1,2,...$} coincides with $L^{2}(0,\pi)$. Thanks.
0
votes
1answer
50 views

Show that H$(I)$ is a closed subspace of $L^2(I)$

EDIT: Original statement is not true, added condition. Let $I$ be the unit interval, define $H(I) = \{f\in AC(I)$ and $f'\in L^2(I)\}$. I want to show that $H(I)$ a closed subspace of $L^2(I)$. ...
5
votes
0answers
132 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
1
vote
1answer
32 views

Looking for a basis of $L^2$ with this special property

The setup. Let $\mathbb{T^2}$ denote the two-dimensional torus, i.e. $$ \mathbb{T}^2 \simeq [-\pi,\pi)^2 $$ induced by identifying opposing faces of $[-\pi,\pi)^2$. Note that $$ L^2(\mathbb{T^2}) ...
1
vote
2answers
99 views

A problem on the bounds of Lp-norms

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ...
3
votes
2answers
101 views

Show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ is not closed in $\ell^2$

How to show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ where $e_n=(\delta_{nk})_{k\in\mathbb N}$ is not closed in $\ell^2$?
2
votes
2answers
118 views

Does $e_n(x)=\exp\left( \frac{i \pi n}{N}x \right)$ define an orthonormal basis of $L^2(-N,N)$?

We know that the Fourier system is complete, i.e. that $\lbrace e_n: ~ n \in \mathbb{N} \rbrace$ defined by \begin{equation} e_n(x)=\frac{1}{\sqrt{2 \pi}}\exp(inx), ~~~ n \in \mathbb{Z} \end{equation} ...
0
votes
1answer
86 views

Is an orthnormal basis of $L^2([0,1])$ also an orthonormal basis of $L^2((0,1))$?

My question is: If $\lbrace e_n \rbrace$ is an orthnormal basis of $L^2([0,1])$, is $\lbrace {e_n}_{|(0,1)} \rbrace$ an orthonormal basis of $L^2((0,1))$? As the points $\lbrace 1 \rbrace$ and ...
4
votes
1answer
132 views

Multiplication operator and trace class

Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators. ...
3
votes
2answers
639 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
1
vote
1answer
748 views

$\ell_p$ is Hilbert space if and only if $p=2$

Can anybody please help me to prove this.. Let p greater than or equal to 1,show that the space of all p-summable sequences is an inner product space if and only if p=2