GATE 2023 MA – Question 56

Functional Analysis · Hilbert Spaces · 2 marks · Numerical answer

In real ℓ² let $M=\{x:\sum_{n\ge1}x_n/4^n=0\}$. The orthogonal projection of (1,1/2,1/3,…) onto M⊥ is $\alpha(\sum_{n\ge1}1/(n4^n))(1/4,1/4^2,1/4^3,\ldots)$. Find α.

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Correct answer: 15 to 15

Explanation

Let v=(4⁻ⁿ). Then M=v⊥ and M⊥=span v. The projection is ⟨h,v⟩v/‖v‖². Here ⟨h,v⟩=Σ1/(n4ⁿ), while ‖v‖²=Σ16⁻ⁿ=1/15. Thus **α=15**.

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