An application of complement functor

I would like to provide an approach for $b)$$b)$ from the homological algebra perspective.

The theorem we will use was proved by Kuang-wei Yang in the paper Completion of Normed Linear Spaces.

Here $N^{\ast }$$N^*$ is the category of normed linear space with contraction maps, and $B^{\ast }$$B^*$ is the category of Banach space with contraction maps.

Now apply the theorem above, let $\begin{array}{c}T:={\int }_0^1(-)dt:C[0,1]\to \mathbb{R}\end{array}$$T:=\int_{0}^1(-)dt:C[0,1]\to\mathbb R\\$, we have the following normal exact sequence

$$0\stackrel{}{\to }P\stackrel{i}{\to }P[0,1]\stackrel{T{|}_{P[0,1]}}{\to }\mathbb{R}\stackrel{}{\to }0$$

Remark. It is easy to see that it is exact, for normal, consider

$$|r{|}_{P[0,1]/P}=inf\{\parallel p{\parallel }_{\mathrm{\infty }}:T(p)=r\}=|r|$$

 

Apply the complement functor, we have the following normal exact sequence

$$0\stackrel{}{\to }A(P)\stackrel{A(i)}{\to }A(P[0,1])\stackrel{A(T{|}_{P[0,1]})}{\to }A(\mathbb{R})\stackrel{}{\to }0$$

Here $A(P[0,1])=C[0,1],A(T{|}_{P[0,1]})=T,A(\mathbb{R})=\mathbb{R}$$A(P[0,1])=C[0,1],A(T|_{P[0,1]})=T,A(\mathbb R)=\mathbb R$, hence $A(P)=V$$A(P)=V$.