# Thread: How to prove that something is an orthonormal basis of L2(N)?

1. ## How to prove that something is an orthonormal basis of L2(N)?

Hi

I am trying to prove the following, but I am getting absolutely nowhere:

{δ_k }_(k=1)^∞ is an orthonormal basis for L^2 (N),where δ_k is the sequence in L^2 (N) whose kth entry is 1.

[/FONT]

2. Originally Posted by Ase
Hi

I am trying to prove the following, but I am getting absolutely nowhere:

{δ_k }_(k=1)^∞ is an orthonormal basis for L^2 (N),where δ_k is the sequence in L^2 (N) whose kth entry is 1.

[/FONT]
and all other entries are 0.

The inner product defined on $\displaystyle L^2(N)$, the space of square summable sequences (also called $\displaystyle l^2$), is $\displaystyle <{a_n},{b_n}>= \sum a_nb_n$ so you need to show three things:

1) That $\displaystyle \{\delta_k\}_{k=1}^\infty$ is a basis for $\displaystyle L^2(N)$

2) That $\displaystyle <\delta_k, \delta_k>= 1$ for all k

3) That $\displaystyle <\delta_k, \delta_j>= 0$ if $\displaystyle k\ne j$

3. $\displaystyle u(i).u(j)=0$ if $\displaystyle i<>j$
$\displaystyle u(i).u(j)=1$ if $\displaystyle i=j$
$\displaystyle ||u(i)||=1$
$\displaystyle x.u(i)=x(i)$
$\displaystyle x.u(i)=0 => x=0$
GTM 233 page 332

4. Thanks a lot HallsofIvy. I will get at it right away and hopefully crack it.

5. ## Basis

Hey,

The sequence

$\displaystyle \{\delta_{k}\}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0, \ldots)$

is also a basis for the normed vector space $\displaystyle l^{1}(\mathbb{N})$.

Couldn't you show that for any $\displaystyle u \in l^{1}(\mathbb{N})$ there exists unique scalars $\displaystyle c_{k} \in \mathbb{C}$ such that

$\displaystyle u=\sum_{k=1}^{\infty} c_{k}\delta_{k}$ ?