Definition:Projection Operator over 2-Sequence Spaces

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Let $\ell^2$ be the $2$-sequence space.

Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space.

Let $\mathbf a \in \ell^2$ be such that:

$\mathbf a = \tuple {a_1, a_2, a_3, \ldots}$

Then by the projection operator over $\ell^2$ we mean the mapping $P_n \in \map {CL} {\ell^2}$ with $n \in \N$ where:

$\tuple {a_1, a_2, a_3, \ldots} \stackrel {P_n} \mapsto \tuple{a_1, a_2, \ldots, a_n, 0, \ldots}$