= Solution
An <orthonormal> <multiresolution analysis> consists of <closed subspaces of a Hilbert space> $(V_j)_{j\in\mathbb Z}$ of the <Hilbert space> $L^2(\mathbb R)$ such that
$$
V_j\subset V_{j+1},\qquad\overline{\bigcup_jV_j}=L^2(\mathbb R),\qquad\bigcap_jV_j=\{0\},\qquad
g\in V_j\ \Longleftrightarrow\ g(2\cdot)\in V_{j+1}.
$$
There is a <scaling function> $\phi$ whose <integer> <function translations> form an <orthonormal basis> of $V_0$. Consequently $\phi_{j,k}(x)=2^{j/2}\phi(2^jx-k)$ form an <orthonormal basis> of $V_j$; in particular $V_0$ is invariant under <integer> <function translations>.
Let $W_j=V_{j+1}\ominus V_j$ be the <orthogonal complement> of $V_j$ in $V_{j+1}$. Nesting, density and the trivial intersection imply the <orthogonal direct sum>
$$
L^2(\mathbb R)=\bigoplus_{j\in\mathbb Z}W_j.
$$
For example, telescoping $V_{J+1}=V_L\oplus\bigoplus_{j=L}^JW_j$ and letting $L\to-\infty$, $J\to\infty$ gives this decomposition; the corresponding <orthogonal projections> converge strongly to zero and the identity by the intersection and density properties.
To construct the <orthonormal wavelet>, use $\phi\in V_1$ to write the <scaling refinement equation> $\phi=\sum_na_n\phi(2\cdot-n)$, with $a\in\ell^2$ and $\sum_n|a_n|^2=2$. Put $m(t)=\frac12\sum_na_ne^{-int}$. The <orthonormal translates and Fourier periodization> criterion proved in part (b), applied to the <scaling refinement equation>, yields
$$
1=\sum_k|\widehat\phi(2t+2\pi k)|^2
=|m(t)|^2\sum_\ell|\widehat\phi(t+2\pi\ell)|^2+|m(t+\pi)|^2\sum_\ell|\widehat\phi(t+\pi+2\pi\ell)|^2
=|m(t)|^2+|m(t+\pi)|^2.
$$
This is the <quadrature mirror filter> identity. Define
$$
q(t)=-e^{-it}\overline{m(t+\pi)},\qquad b_n=(-1)^n\overline{a_{1-n}},\qquad
\psi(x)=\sum_nb_n\phi(2x-n).
$$
The <Fourier series> of $q$ is $\frac12\sum_nb_ne^{-int}$, and $\widehat\psi(2t)=q(t)\widehat\phi(t)$. The two rows $(m(t),m(t+\pi))$ and $(q(t),q(t+\pi))$ form a unitary $2\times2$ <matrix> almost everywhere: both have squared <norm> one and their <inner product> is zero. This <wavelet completion of a multiresolution filter> proves not just orthogonality but completeness. Splitting a finer-scale coefficient <Fourier series> into its values at $t$ and $t+\pi$, this invertible two-channel matrix resolves it into a coarse-scale channel and its complementary channel. Thus the <integer> <function translations> of $\psi$ form an <orthonormal basis> of $W_0$.
The dilated <functions> $\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k)$ therefore form an <orthonormal basis> of each $W_j$. The <orthogonal direct sum> proves the conclusion: \b[every <orthonormal> multiresolution analysis produces an <orthonormal> wavelet <basis>] $\{\psi_{j,k}:j,k\in\mathbb Z\}$ of $L^2(\mathbb R)$.
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