3
votes
1answer
43 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
2
votes
1answer
139 views

$\ell^{\infty}(\mathbb N)$ is not a separable space

I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable. My attempt Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
2
votes
0answers
185 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
1
vote
1answer
72 views

Star graph embeddings

This is an homework question which I'm struggling with: Let $S = (V, E, w)$ a star graph, meaning, $S$ is a tree that all it's vertices are leafs except one. I need to : show that every weighted ...
0
votes
1answer
123 views

Isometric embedding of an $n$-point equilateral space

I'm stumped on these questions, and would appreciate a solution: I need to find an isometric embedding of the n-point equilateral space in $l_{p}$. And if $n=2^{d}$, an isometric embedding of the ...
1
vote
1answer
48 views

Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property

$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $. To prove ...
7
votes
2answers
158 views

On $L^p$ and $\ell^p$

If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
3
votes
1answer
133 views

Closure of $l_1$ in $l_\infty$

Suppose we have a set $A$ which is the set of all sequences that satisfy $|x_n|\xrightarrow{} 0$. If we consider $l_1$ to be a subset of $l_\infty$. Show that the closure of $l_1$ in $l_\infty$ equals ...