Hilbert Sequence Space is Arc-Connected

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Theorem

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.


Then $\ell^2$ is arc-connected.


Proof

Let $x = \sequence {x_i}$ and $y = \sequence {y_i}$.

Consider the mapping $f: \closedint 0 1 \to \ell^2$ defined as:

$\forall t \in \closedint 0 1: \map f t = t x + \paren {1 - t} y = \sequence {t x_i + \paren {1 - t} y_i}$


\(\ds \sum_{i \mathop \ge 0} \paren {t x_i + \paren {1 - t} y_i}^2\) \(=\) \(\ds \sum_{i \mathop \ge 0} \paren {t^2 x_i^2 + \paren {1 - t}^2 y_i^2 + 2 t \paren {1 - t} x_i y_i}\)
\(\ds \) \(=\) \(\ds t^2 \sum_{i \mathop \ge 0} x_i^2 + \paren {1 - t}^2 \sum_{i \mathop \ge 0} y_i^2 + 2 t \paren {1 - t} \sum_{i \mathop \ge 0} x_i y_i\)

which is convergent.


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Then $f$ is an injective path joining $x$ to $y$.

Hence the result.

$\blacksquare$


Sources

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