Corollary 4.3 If Condition 4.1 holds then B$_{K,w}$ is empirically identifiable wpa1. Condition 4.1 is therefore a sufficient condition for (30) to hold for the linear sieve space B$_{K,w}$ . Remark 4.1 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 . (1) For i.i.d. regressors (and λ$_{K,n}$ ≲ 1 ), previously Huang (1998) establishes equivalence of the theoretical and empirical L 2 norms over the sieve space via a chaining argument with ζ 2 $_{K,n}$K/n = o (1) . Huang (2003b) relaxes this to K (log n ) /n = o (1) for a polynomial spline basis. Our Lemma 2.1 shows that, in fact, ζ$_{K,n}$ √ (log K ) /n = o (1) is sufficient with an arbitrary linear sieve (provided λ$_{K,n}$ ≲ 1 ). (2) For strictly stationary beta-mixing regressors (and λ$_{K,n}$ ≲ 1 ), Lemma 2.2 shows the equivalence of the theoretical and empirical L 2 norms over any linear sieve space under either ζ$_{K,n}$ √ (log n ) $^{2}$/n = o (1) for exponential beta-mixing, or ζ$_{K,n}$ √ (log K ) /n γ/ (1+ γ ) = o (1) for algebraic beta-mixing. In this section we show that the L $^{2}$( X ) orthogonal projection onto (tensor product) compactly sup- ported wavelet bases is stable in sup norm as the dimension of the space increases. Consider the orthogonal projection operator P$_{K}$ defined in expression (16) where the elements of b K span the tensor products of d univariate wavelet spaces Wav( K$_{0}$, [0 , 1]). We show that its L ∞ operator norm ‖ P$_{K}$ ‖ ∞ (see expression (17)) is stable, in the sense that ‖ P$_{K}$ ‖ ∞ ≲ 1 as K → ∞ . We also show that the empirical L 2 projection P$_{K,n}$ onto the wavelet sieve is stable in sup norm wpa1. This result is used to establish that series LS estimators with (tensor-product) wavelet bases attain their optimal sup-norm rates. A variant of this result for projections arising in series two-stage LS was used in an antecedent of this paper (Chen and Christensen, 2013) but its proof was omitted for brevity. The following Theorem presents our result for the stability of the projection with respect to the L $^{2}$( X ) inner product. Theorem 5.1 Let X ⊇ [0 , 1] d and let the density f$_{X}$ of X$_{i}$ be such that 0 < inf$_{x}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ d f$_{X}$ ( x ) ≤ sup$_{x}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ d f$_{X}$ ( x ) < ∞ . Let B$_{K}$ be the tensor product of d univariate wavelet spaces Wav( K$_{0}$, [0 , 1]) where Wav( K$_{0}$, [0 , 1]) is as described in Section 6 and K = 2 dJ and K$_{0}$ = 2 J > 2 N . Then: ‖ P$_{K}$ ‖ ∞ ≲ 1 . 28 5 Sup-norm stability of L $^{2}$( X ) projection onto wavelet sieves