Part BCSIR NET June 2025the-sin-squared-weight-and-the-2-over-pi-make-these-orthonormal-not-merely-orthogonal

The sin squared weight and the 2 over pi make these orthonormal not merely orthogonal

Let $C [0, π]$ be the real vector space of real-valued continuous functions on the closed interval $[0, π]$ . For positive integers n, define $f_{n} \in C [0, π]$ by $f_{n} (x) = sin ($ nx)/sin x if $x \in (0, π), f_{n} (0) = n$ , and $f_{n} (π) = (- 1)^{n}$ ⁻ $^{1} n$ . Let V be the real subspace of $C [0, π]$ spanned by { $f_{1}, f_{2}, f_{3}$ }. Consider the inner product on V given by ⟨f, g⟩ $= (2/ π) \int_{0}^{π} f (x) g (x) sin^{2} x$ dx. Which of the following statements is true?

A. $f_{4} \in V$
B.{ $f_{1}, f_{2}, f_{3}$ } is an orthonormal basis of V.✓
C.The dimension of V is 2.
D.{ $f_{1}, f_{2}, f_{3}$ } is an orthogonal set but not orthonormal.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

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The sin squared weight and the 2 over pi make these orthonormal not merely orthogonal

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