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.
Thanks in advance
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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.
Thanks in advance
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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 $\displaystyle k\ne j$
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
$\displaystyle u=\sum_{k=1}^{\infty} c_{k}\delta_{k}$ ?