= Solution
For a <pathwise differentiable statistical functional>, an <influence-function representer> is a <mean-zero function> $\varphi\in L^2(P)$ such that every admissible <score function> $g$ satisfies $D\psi_P(g)=P(\varphi g)$. The <efficient influence function>, also called the <canonical gradient>, is the unique such representer in the <statistical tangent space>. Equivalently, it is the <orthogonal projection> of any representer onto that <statistical tangent space>. The <Pythagorean theorem in an inner-product space> shows that it has the smallest squared <L2 norm> among all representers.
Here the <statistical tangent space> is all of $L^2_0(P_f)$. To verify the closure explicitly, take $h\in L^2_0(P_f)$, truncate it to $h_m=\max(-m,\min(h,m))$, and set $g_m=h_m-P_fh_m$. Then $g_m$ is bounded and centered, and $g_m\to h$ in $L^2(P_f)$, by <dominated convergence> and the <Cauchy-Schwarz inequality>. Part (c) supplies a representer already in this space. Hence
$$
\boxed{\varphi_f(u)=a(u)-\int_0^1 a(v)f(v)\,dv.}
$$
Its <variance> is $P_fa^2-(P_fa)^2$.
\b[The closure must be taken in the density-weighted space $L^2(P_f)$.] An unweighted reading of $L^2[0,1]$ in the printed hint is false. For example, when $f(u)=2u$, the <function> $h(u)=u^{-3/4}-8/5$ has $P_fh=0$ and $P_fh^2=36/25$, so bounded centered truncations converge to it in $L^2(P_f)$; nevertheless $h\notin L^2([0,1],du)$. This illustrates <density of bounded centered scores> and fixes the measure in the closure statement.
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