= Solution
Put $W_j=V_{j+1}\cap V_j^\perp$, the detail space of the <multiresolution analysis>. The <Meyer-Mallat theorem> gives a function $\psi\in W_0$ whose integer translates form an <orthonormal basis> of $W_0$. Dilations then give an <orthonormal basis> of each $W_j$, and the density and trivial-intersection axioms imply
$$
\boxed{L^2(\mathbb R)=\mathop{\bigoplus}_{j\in\mathbb Z}W_j.}
$$
For an explicit filter construction, write the <scaling refinement equation> $\varphi(x)=\sqrt2\sum_kh_k\varphi(2x-k)$. With a <quadrature mirror filter> one may take $g_k=(-1)^k\overline{h_{1-k}}$ and $\psi(x)=\sqrt2\sum_kg_k\varphi(2x-k)$. A harmless constant sign or integer translation gives an equivalent <orthonormal wavelet>.
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