List of top Functional Analysis Questions on Bounded Linear Operators on Sequence Spaces asked in GATE MA

Let \(\ell^{\infty} = \{x = (x_n)_{n \geq 1} \mid x_n \in \mathbb{R},\ \sup\{|x_n| : n = 1, 2, \ldots\} < \infty\}\) with the supremum norm. Let \(T : \ell^{\infty} \to \ell^{\infty}\) be given by \(T(x_1, x_2, x_3, \ldots) = \left(x_1, \dfrac{x_2}{2}, \dfrac{x_3}{3}, \ldots \right)\). Which one of the following is TRUE?