Teaching JKPSC Lecturership Mock Test Series 2024-25 Mathematical Science Linear Algebra Inner Product Spaces, Orthonormal Basis
Let L2[-1, 1] be the Hilbert space of real valued square integrable functions on [-1, 1] equipped with the norm \(\|f\|=\left(\int_{-1}^1|f(x)|^2 d x\right)^{1 / 2}\).
Consider the subspace M = {f ∈ L2[-1, 1] : \(\int_{-1}^1 f(x) d x=0\)}.
For f(x) = x2, define d = inf {||f - g|| : g ∈ M}. Then
1
\(d=\frac{\sqrt{2}}{3}\)
2
\(d=\frac{2}{3}\)
3
\(d=\frac{3}{\sqrt{2}}\)
4
\(d=\frac{3}{2}\)