Question

Let \( L^2[-1, 1] \) be the Hilbert space of real-valued square integrable functions on [-1, 1] equipped with the norm \[ \|f\| = \left( \int_{-1}^{1} |f(x)|^2 \, dx \right)^{1/2}. \] Consider the subspace \[ M = \{ f \in L^2[-1, 1] : \int_{-1}^{1} f(x) \, dx = 0 \}. \] For \( f(x) = x^2 \), define \[ d = \inf \{ \|f - g\| : g \in M \}. \] Then

• A. \( d = \sqrt{3}/3 \)
• B. \( d = 2/3 \)
• C. \( d = 3/\sqrt{2} \)
• D. \( d = 3/2 \)

Hint:

When working with Hilbert spaces, use orthogonal projections to minimize the distance to a subspace.

Answer 1

The Correct Option is A

The problem asks us to find the distance of \( f(x) = x^2 \) from the subspace \( M \). We use the orthogonal projection onto \( M \) to minimize the distance. The orthogonal projection of \( f(x) \) onto \( M \) is the function whose integral over the interval \( [-1, 1] \) is zero. Using standard methods in functional analysis, the result gives the distance \( d = \sqrt{3}/3 \). Final Answer: (A) \( d = \sqrt{3}/3 \).