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
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations