e.g., Newey (1997), Huang (1998), and Chen (2007)). See DeVore and Lorentz (1993) and Belloni et al. (2014) for additional bases with either ζ$_{K,n}$ ≲ √$_{K}$ or ζ$_{K,n}$ ≲ K properties. Let ˜ b K $_{w}$( x ) denote the orthonormalized vector of basis functions, namely ˜ b K $_{w}$( x ) = E [ b K $_{w}$( X$_{i}$ ) b K $_{w}$( X$_{i}$ ) $^{′}$]− 1 / $^{2}$b K $_{w}$( x ) , (11) Assumption 5 Either: (a) { X$_{i}$ } n i =1 is i.i.d. and ζ$_{K,n}$λ$_{K,n}$ √ (log K ) /n = o (1) , or (b) ‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − I$_{K}$ ‖ = o$_{p}$ (1) . and let ˜ B$_{w}$ = ( ˜ b K $_{w}$( X$_{1}$ ) , . . . , ˜ b K $_{w}$( X$_{n}$ )) $^{′}$. Assumption 5 is a mild but powerful condition that ensures the empirical and theoretical L 2 norms are equivalent over the linear sieve space wpa1 (see Section 4 for details). In fact, to establish many of our results below with weakly dependent data, nothing further about the weak dependence properties of the regressor process { X$_{i}$ } n i =1 needs to be assumed beyond convergence of ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ to zero. In the i.i.d. case, the following Lemma shows that part (a) of Assumption 5 automatically implies ‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − I$_{K}$ ‖ = o$_{p}$ (1). Lemma 2.1 Under Assumption 4(iii), if { X$_{i}$ } n i =1 is i.i.d. then provided ζ$_{K,n}$λ$_{K,n}$ √ (log K ) /n = o (1) . Remark 2.2 Consider the compact support case in which X = [0 , 1] d and w$_{n}$ ( x ) = 1 for all x ∈ X and all n (so that b$_{Kk}$ ( x ) w$_{n}$ ( x ) = b$_{Kk}$ ( x ) for all n and K ) and suppose the density of X$_{i}$ is uniformly bounded away from zero and infinity over X . In this setting, we have λ$_{K,n}$ ≲ 1 . If { X$_{i}$ } n i =1 is i.i.d., then Assumption 5 is satisfied with √ ( K log K ) /n = o (1) for spline, trigonometric polynomial or wavelet bases, and with K √ (log K ) /n = o (1) for (tensor-product) power series. When the regressors are β -mixing (see Section 4 for definition), the following Lemma shows that Assumption 5(b) is still easily satisfied. ‖ ( ˜ B ′ w ˜ B$_{w}$/n ) − I$_{K}$ ‖ = O$_{p}$ ( ζ$_{K,n}$λ$_{K,n}$ √ (log K ) /n ) = o$_{p}$ (1) 10