Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space
Theorem
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the 2-sequence space equipped with Euclidean norm.
Let $c_{00}$ be the space of almost-zero sequences.
Then $c_{00}$ is everywhere dense in $\struct {\ell^2, \norm {\, \cdot \,}_2}$
Proof
Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^2$.
By definition of $\ell^2$:
- $\ds \sum_{i \mathop = 0}^\infty \size {x_i}^2 < \infty$
Let $\ds s_n := \sum_{i \mathop = 0}^n \size {x_i}^2$ be a sequence of partial sums of $\ds s = \sum_{i \mathop = 0}^\infty \size {x_i}^2$.
We have that $s$ is a convergent sequence:
- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies \size {s_n - s} < \epsilon$
Note that:
\(\ds \size {s_n - s}\) | \(=\) | \(\ds \size {\sum_{i \mathop = 0}^n \size {x_i}^2 - \sum_{i \mathop = 0}^\infty \size {x_i}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \size {\sum_{i \mathop = n \mathop + 1}^\infty \size {x_i}^2}\) |
Let $\ds N \in \N : \sum_{n \mathop = N + 1}^\infty \size {x_n}^2 < \epsilon^2$
Let $\mathbf y := \tuple {x_1 \ldots, x_N, 0, \ldots}$.
By definition, $\mathbf y \in c_{00}$.
We have that:
\(\ds \norm {\mathbf x - \mathbf y}_2^2\) | \(=\) | \(\ds \sum_{i \mathop = 0}^\infty \size {x_i - y_i}^2\) | Definition of Euclidean Norm | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = N + 1}^\infty \size {x_i}^2\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm {\mathbf x - \mathbf y}_2\) | \(<\) | \(\ds \epsilon\) |
Hence by definition, $c_{00}$ is dense in $\ell^2$.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces