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GATE 2020 MA – Question 36

Functional Analysis · Hilbert Spaces · 2 marks · Multiple choice

Let $(e_n)_{n\geq1}$ be an orthonormal basis of a separable Hilbert space H and set $f_n=e_n-e_{n+1}/(n+1)$. Which is true?

  1. The closure of the span of all fₙ is H.
  2. If $\langle f,f_n\rangle=\langle f,e_n\rangle$ for all n, then f = 0.
  3. The set of all fₙ is orthogonal.
  4. There is no nonzero f with $\langle f,e_2\rangle=\langle f,f_2\rangle$.

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Correct answer: (A) The closure of the span of all fₙ is H.

Explanation

If a vector is orthogonal to every fₙ, its coefficients obey $a_{n+1}=(n+1)a_n$, hence $a_n=n!a_1$. Square summability forces a₁ = 0, so all coefficients vanish. The orthogonal complement of the span is therefore zero and the span is dense, proving A. For B and D, f = e₁ is a counterexample. Consecutive fₙ also have nonzero inner product, so C fails.

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