= Solution
The <linear independence> of the first $n$ elements of each <orthonormal system> shows that $T_n$ and $S_n$ have the same finite <dimension> $n$ and are <closed subspaces of a Hilbert space>. Apply part (b) with $U=T_n$ and $V=S_n^\perp$. Since $(S_n^\perp)^\perp=S_n$, the positive <directed subspace angle> cosine gives
$$
H=T_n\oplus S_n^\perp.
$$
Let $Q$ be the <oblique projection> onto $T_n$ along $S_n^\perp$. Define $\widetilde f_n=Qf$. Its residual $f-Qf$ is orthogonal to every $\psi_j$ with $1\le j\le n$, so the required measurements agree. Conversely, if $g\in T_n$ has the same measurements, then $f-g\in S_n^\perp$; uniqueness of the <direct sum> decomposition gives $g=Qf$. Thus this is <finite-dimensional Hilbert sampling reconstruction>.
There is also an explicit coefficient description. Use the <inner product> convention linear in its first entry and put
$$
C_{jk}=\langle\phi_k,\psi_j\rangle,\qquad b_j=\langle f,\psi_j\rangle,\qquad\widetilde f_n=\sum_{k=1}^n c_k\phi_k.
$$
Then $Cc=b$. The <orthonormal systems> show $\|P_{S_n}\sum_k c_k\phi_k\|=\|Cc\|_2$ and $\|\sum_k c_k\phi_k\|=\|c\|_2$, so the smallest <singular value> of $C$ is $\cos\theta_{T_n,S_n}>0$. Hence $c=C^{-1}b$, another direct proof of existence and uniqueness.
For $n\ge1$, both summands of this <direct sum> are nonzero: the infinite <orthonormal system> $(\psi_j)$ contains $\psi_{n+1}\in S_n^\perp$. The permitted <oblique projection> norm formula therefore applies without its degenerate exception, giving
$$
\|Q\|=\|I-Q\|=\sec\theta_{T_n,S_n}.
$$
The <operator norm> immediately yields the stability estimate $\|\widetilde f_n\|\le\sec\theta_{T_n,S_n}\|f\|$. For the approximation bounds, put $e=f-P_{T_n}f$. Because $Q$ fixes $T_n$, we have
$$
f-Qf=(I-Q)e,
$$
so the <operator norm> estimate gives $\|f-Qf\|\le\sec\theta_{T_n,S_n}\|e\|$. For the lower bound, $e\perp T_n$ while $P_{T_n}f-Qf\in T_n$. The <Pythagorean identity> gives
$$
\|f-Qf\|^2=\|e\|^2+\|P_{T_n}f-Qf\|^2\ge\|e\|^2.
$$
\b[The unique measurement-matching reconstruction is stable and within the <secant function> factor of the best <orthogonal projection> approximation:]
$$
\boxed{\widetilde f_n=Qf,\quad\|\widetilde f_n\|\le\sec\theta_{T_n,S_n}\|f\|,\quad\|f-P_{T_n}f\|\le\|f-\widetilde f_n\|\le\sec\theta_{T_n,S_n}\|f-P_{T_n}f\|.}
$$
If a zero-dimensional reconstruction is admitted, it is simply $\widetilde f_0=0$ and its error equals the <norm> of $f$; that case is best stated directly instead of using the angle of a zero space.
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