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The canonical gradient is the influence-function representer in the statistical tangent space $\mathcal{T}$. It equals the orthogonal projection onto $\mathcal{T}$ of any representer. Every other representer differs from it by an element of ${\mathcal{T}}^{\perp }$, so the Pythagorean theorem in an inner-product space gives it minimum variance. For $\psi (P)=Pa$ in an unrestricted density model with bounded $a$, it is $a-Pa$.

An influence-function representer is a centered square-integrable function representing derivatives of a statistical functional along all admissible score functions. Representers may differ by a function orthogonal to the statistical tangent space. This pathwise definition is distinct from defining an influence function solely by point-mass contamination paths.

Choose a collection of statistical paths through $P_f$ that are differentiable in quadratic mean. A statistical tangent set at $P_f$ is the set of their score functions. In particular each member belongs to $L_0^2(P_f)$, by the mean-zero score identity under quadratic-mean differentiability.

A statistical tangent set records which first-order directions the chosen statistical paths can realize. Its closed linear span in L2 space is the statistical tangent space. The tangent set consists of attainable scores; the tangent space also includes their linear combinations and $L^2(P_f)$ limits.

For a pathwise differentiable statistical functional, an influence-function representer is a mean-zero function $\phi \in L^2(P)$ such that every admissible score function $g$ satisfies $D{\psi }_P(g)=P(\phi g)$. 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 $L_0^2(P_f)$. To verify the closure explicitly, take $h\in L_0^2(P_f)$, truncate it to $h_m=\max (-m,\min (h,m))$, and set $g_m=h_m-P_f{h}_m$. 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. HenceIts variance is $P_f{a}^2-(P_{f}a{)}^2$.

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 $P_{f}h=0$ and $P_f{h}^2=36/25$, 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.

A statistical functional is pathwise differentiable relative to chosen statistical paths if its derivative along every path depends only on that path's score function and defines a bounded linear functional on their statistical tangent space. The Riesz representation theorem expresses this derivative as an L2 inner product with a unique element of the statistical tangent space, the canonical gradient. A path family and the associated derivative remainder conditions must both be specified; a formal derivative along one convenient family does not establish differentiability along all paths.