= Solution
An <orthonormal wavelet> is a function whose integer translates and dyadic dilates
$$
\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k),\qquad j,k\in\mathbb Z,
$$
form an <orthonormal basis> of the <Hilbert space> $L^2(\mathbb R)$. Thus their pairwise <inner products> are $\delta_{j,j'}\delta_{k,k'}$, and their closed linear span is the whole space. Equivalently, every $f\in L^2(\mathbb R)$ has the <norm>-convergent expansion
$$
\boxed{f=\sum_{j,k}\langle f,\psi_{j,k}\rangle\psi_{j,k},\qquad \|f\|_2^2=\sum_{j,k}|\langle f,\psi_{j,k}\rangle|^2.}
$$
The factor $2^{j/2}$ preserves the $L^2$ <norm> under dilation.
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