Left Shift Operator on 2-Sequence Space is Continuous
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Theorem
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $2$-sequence space with $2$-norm.
Let $L : \ell^2 \to \ell^2$ be the left shift operator.
Then $L$ is continuous on $\struct {\ell^2, \norm {\, \cdot \,}_2}$.
Proof
Let $\sequence {a_n}_{n \mathop \in \N} = \tuple {a_1, a_2, a_3, \ldots}$ be a $2$-sequence.
\(\ds \norm {\map L {\sequence {a_n}_{n \mathop \in \N} } }_2\) | \(=\) | \(\ds \norm { L \tuple {a_1, a_2, a_3, \ldots} }_2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \norm {\tuple {a_2, a_3, a_4, \ldots} }_2\) | Definition of Left Shift Operator | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\sum_{i \mathop = 2}^\infty \size {a_i}^2}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sqrt {\sum_{i \mathop = 1}^\infty \size {a_i}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \cdot \norm {\sequence {a_n}_{n \mathop \in \N} }_2\) |
By Continuity of Linear Transformation between Normed Vector Spaces, $L$ is continuous in $\struct {\ell^2, \norm {\, \cdot \,}_2}$.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$