Show that $\operatorname{span}\{e_i\}$ is dense in $\ell^p$.

functional-analysishilbert-spaces

I'm new to Hilbert Spaces, especially to the $\ell^p$ spaces. I want to proof why $\operatorname{span}\{e_i\}$ is dense in $\ell^p$ for $ 1 \leq p < + \infty$.
To show that $\operatorname{span}\{e_i\}$ is dense in $\ell^p$, I consider a sequence in $\ell^p$ and then show that $e_k = (0,…,0,1,0,…,0)$ lies in an open neighborhood of $u_k$.

Since $(u_n)_n$ is in $\ell^p$, we know that
$\left(\sum_{k \in \mathbb{N}} |u_k|^p\right)^{\frac{1}{p}} \leq + \infty $ .

So the series converges, there exits a rank $N \in \mathbb{N}$, such that
$$ \sum_{k \geq N} |u_k|^p \leq \epsilon^p
$$

Now for that $e_k$ lies in a neighborhood of $u_k$, we want to show
$\|u – e\|_{\ell^p} \leq \epsilon'$.

Now,
$\sum_{k \in \mathbb{N}}|u_k – e_k|$.

What now? I would also appreciate improvements to my mathematical expressions.
Thank you!

EDIT:
My attempt after reading you suggestions (I will write the proof in
detail to be sure I have understood everything correctly):

Consider a sequence $(u_n)_n$ in $\ell^p$. We then know that $\left(\sum_{k \in \mathbb{N}} |u_k|^p\right)^{\frac{1}{p}} \leq + \infty $, thus the sequence converges and $\forall \epsilon > 0$ there is a rank $N \in \mathbb{N}$ such that,
$$ \sum_{k \geq N} |u_k|^p \leq \epsilon^p
$$
Consider $v = u_1e_1 + … + u_{N-1}e_{N-1}$ which clearly is in $\operatorname{span}\{e_i\}$ as a linear combination of the vectors $e_k$.
$v$ is of the form $(u_1, …, u_{N-1},0,….)$ (a sequence that is zero at a certain rank).
Now compute $\| u – v \|_{\ell^p}$:
\begin{align} \| u – v \|_{\ell^p}^p &= \sum_{k \in \mathbb{N}} | u_k -v_k |^p \\ &= | u_0 -v_0 |^p + | u_1 – v_1 |^p + \dots + |u_{N-1} – v_{N-1}|^p + |u_N – v_N |^p + \dots \\
& = \sum_{k \geq N} |u_k – v_k|^p \\
&= \sum_{k \geq N} |u_k|^p \leq \epsilon ^p
\end{align}
Thus $$
\| u – v \|_{\ell^p} \leq \epsilon
$$
We have shown that $v$ lies in a neighborhood of $u$ and conclude that $\operatorname{span}\{e_i\}$ is dense in $\ell^p$.

Best Answer

Errors:

  1. $\ell^{p}$ consists of infinite sequences so $e_k=(0,0,...,1,0,0...)$ with $1$ in the $k-$th place. [It is not a finite sequence]

  2. $u_k$ is scalar and $e_k$ is a vector so $u_k-e_k$ makes no sense.

You have started the proof correctly by obtaining $N$ using $\epsilon$. Now consider $v=u_1e_1+u_2e_2+\cdots+u_{N-1}e_{N-1}\equiv (u_1,u_2,\cdots,u_{N-1},0,0,\cdots)$. To complete the proof estimate $\|v-u\|$.

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