# a linear map on $W$

Define $$W = \{(a_1, a_2,\cdots) : a_i \in \mathbb{F}, \exists N\in\mathbb{N}, \forall n \geq N, a_n = 0\},$$ where $$\mathbb{F} = \mathbb{R}$$ or $$\mathbb{C}$$ and $$W$$ has the standard inner product, which is given by $$\langle(a_1,a_2,\cdots), (b_1,b_2,\cdots)\rangle = \sum_{i=1}^\infty a_i \overline{b_i}$$ ($$\overline{b_i}$$ is simply the complex conjugate of $$b_i$$). Prove that the linear map $$T : W \to W$$ given by $$T(a)_j = \sum_{i=j}^\infty a_i$$, where $$T(a) = (T(a)_1, T(a)_2,\cdots),$$ has no adjoint.

I know that to show that $$T$$ has an adjoint $$T^*$$, it suffices to show that for all $$a,b \in W$$, $$\langle T(a),b \rangle = \langle a, T^* b\rangle$$. So to show that $$T$$ does not have an adjoint, it suffices to show that there is no linear map $$T^*$$ so that for all $$a,b \in W \langle T(a),b\rangle = \langle a,T^*b\rangle$$ For any $$a \in W,$$ we may find $$N\in\mathbb{N}$$ st $$i \geq N\Rightarrow a_i = 0.$$ Hence $$\langle T(a),b\rangle = \sum_{i=1}^\infty \left(\sum_{j=i}^\infty a_j\right) \overline{b_i}=\sum_{i=1}^{N-1} \left(\sum_{j=i}^{N-1} a_j\right)\overline{b_i}.$$ Also, $$\langle a, T^*b\rangle = \sum_{i=1}^\infty a_i\overline{(T^*b)_i}=\sum_{i=1}^{N-1} a_i\overline{(T^*b)_i}.$$ Hence $$\langle T(a),b\rangle = \langle a,T^*b\rangle \iff \sum_{i=1}^{N-1} \left(\sum_{j=1}^{N-1} a_j \overline{b_i}-a_i \overline{(T^*b)_i}\right) = 0.$$ I know I'm supposed to find a $$b \in W$$ that'll make it impossible for $$\langle a,T^*b\rangle = \langle T(a),b\rangle$$ for all $$a\in W$$, but I'm unsure how to find this.

Let $$e_i\in W$$ satisfy $$(e_i)_j=1$$ if $$i=j$$ and $$(e_i)_j=0$$ otherwise. Then $$\langle Te_i,e_j\rangle=\cases{1\,\,{\rm if}\,\,j\leq i,\\0 \,\,{\rm if}\,\,j>i.}$$
Thus if $$T^*$$ exists: $$\langle e_i,T^*e_j\rangle=\cases{1\,\,{\rm if}\,\,j\leq i,\\0 \,\,{\rm if}\,\,j>i.}$$
So we have $$(T^*e_j)_i=1$$ for all $$i\geq j$$ which contradicts $$T^*e_j\in W$$.
Suppose that $$T^*$$ exists. Exchanging the sum (no issues because every sequence is finite), $$\langle T(a),b\rangle = \sum_{i=1}^\infty (\sum_{j=i}^\infty a_j) \overline{b_i}=\sum_{j=1}^\infty\sum_{i=1}^ja_j\overline{b_j}=\sum_{j=1}^\infty a_j\overline{\sum_{i=1}^j{b_j}}.$$ So $$T^*(b)_j=\sum_{i=1}^j{b_j}.$$ But then $$T^*(b)\not\in W$$ for any nonzero $$b$$ (as it would have infinitely many nonzero entries). So $$T^*$$ does not exist.