Let \(\ell ^1\) be the Banach space of real sequences \(a=(a_1,a_2,\ldots )\) with norm \[ \lVert a\rVert _1=\sum _{i=1}^{\infty }|a_i|<\infty , \] and \(\ell ^\infty \) the Banach space of real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\). Let \(a\in \ell ^1\) be a fixed sequence; for any sequence \(x\in \ell ^\infty \), define a new sequence \(T_a(x)\) by \[ \bigl (T_a(x)\bigr )_n=\sum _{i=1}^{n}a_i x_i \]
(a) Show that this defines a bounded operator \(T_a:\ell ^\infty \to \ell ^\infty \).
(b) Show \(\lVert T_a\rVert =\lVert a\rVert _1\), where the left-hand side denotes the norm of the operator \(T_a:\ell ^\infty \to \ell ^\infty \).