Infinite Limit Operator is Linear Mapping

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Theorem

Let $c$ be the space of convergent sequences.

Let $\R$ be the set of real numbers.

Let $L : c \to \R$ be the infinite limit operator.


Then $L$ is a linear mapping.


Proof

Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N}, \mathbf y = \sequence {y_n}_{n \mathop \in \N} \in c$.

Suppose $\mathbf x$ and $\mathbf y$ converge to $x$ and $y$ respectively.

Let $\alpha \in \R$.


Distributivity

\(\ds \map L {\mathbf x + \mathbf y}\) \(=\) \(\ds \map {\lim_{n \mathop \to \infty} } {x_n + y_n}\) Definition of Infinite Limit Operator
\(\ds \) \(=\) \(\ds x + y\) Sum Rule for Real Sequences
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} x_n + \lim_{n \mathop \to \infty} y_n\)
\(\ds \) \(=\) \(\ds \map L {\mathbf x} + \map L {\mathbf y}\) Definition of Infinite Limit Operator

$\Box$


Positive homogenity

\(\ds \map L {\alpha \mathbf x}\) \(=\) \(\ds \map {\lim_{n \mathop \to \infty} } {\alpha x_n}\)
\(\ds \) \(=\) \(\ds \alpha x\) Multiple Rule for Real Sequences
\(\ds \) \(=\) \(\ds \alpha \lim_{n \mathop \to \infty} {x_n}\)
\(\ds \) \(=\) \(\ds \alpha \map L {\mathbf x}\) Definition of Infinite Limit Operator

$\blacksquare$


Sources

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