Left Shift Operator is Linear Mapping

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Theorem

Let $X = Y = \ell^2$ be 2-sequence spaces over real numbers.

Let $L : X \to Y$ be the left shift operator.


Then $L$ is a linear mapping.


Proof

Let $x = \tuple {x_1, x_2,x_3, \ldots}, y = \tuple {y_1, y_2, y_3, \ldots} \in \ell^2$

Let $\alpha \in \R$.


Distributivity

\(\ds \map L {x + y}\) \(=\) \(\ds \map L {\tuple {x_1, x_2,x_3, \ldots} + \tuple {y_1, y_2, y_3, \ldots} }\)
\(\ds \) \(=\) \(\ds \map L {\tuple {x_1 + y_1, x_2 + y_2, x_3 + y_3, \ldots} }\) P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space
\(\ds \) \(=\) \(\ds \tuple {x_2 + y_2, x_3 + y_3, \ldots}\) Definition of Left Shift Operator
\(\ds \) \(=\) \(\ds \tuple {x_2, x_3, \ldots} + \tuple {y_2, y_3, \ldots}\) P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space
\(\ds \) \(=\) \(\ds \map L {\tuple {x_1, x_2, x_3, \ldots} } + \map L {\tuple {y_1, y_2, y_3, \ldots} }\) Definition of Left Shift Operator
\(\ds \) \(=\) \(\ds \map L x + \map L y\)

$\Box$


Positive homogenity

\(\ds \map L {\alpha x}\) \(=\) \(\ds \map L {\alpha \tuple {x_1, x_2, x_3, \ldots} }\)
\(\ds \) \(=\) \(\ds \map L {\tuple {\alpha x_1, \alpha x_2, \alpha x_3, \ldots} }\) P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space
\(\ds \) \(=\) \(\ds \tuple {\alpha x_2, \alpha x_3, \ldots}\) Definition of Left Shift Operator
\(\ds \) \(=\) \(\ds \alpha \tuple {x_2, x_3, \ldots}\) P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space
\(\ds \) \(=\) \(\ds \alpha \map L {\tuple {x_1, x_2, x_3, \ldots} }\) Definition of Left Shift Operator
\(\ds \) \(=\) \(\ds \alpha \map L x\)

$\blacksquare$


Sources

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