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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.

Let $g$ be a bounded function with $P_{f}g=0$. A bounded density tilt realizes this direction:These are nonnegative probability density functions, since $\int f_t=1+t{P}_{f}g=1$. The uniform Taylor expansion of the square root givesThe squared L2 norm of this remainder is $O(t^4{P}_f{g}^4)=o(t^2)$, since $g$ is bounded. Thus the statistical path is differentiable in quadratic mean with score function $g$.

Conversely every score function is centered by the mean-zero score identity under quadratic-mean differentiability. Choosing all these bounded density tilts therefore gives the statistical tangent setThis is a valid choice of statistical tangent set; it does not assert that every possible score function in the unrestricted density model is bounded.