Question

Let \( X \) be the normed space \( ({R}^2, \|\cdot\|) \), where \[ \| (x, y) \| = |x| + |y|, \quad (x, y) \in {R}^2. \] Let \( S = \{ (x, 0) : x \in {R \} \) and \( f : S \to {R} \) be given by \( f((x, 0)) = 2x \) for all \( x \in {R} \). Recall that a Hahn–Banach extension of \( f \) to \( X \) is a continuous linear functional \( F \) on \( X \) such that \( F|_S = f \) and \( \|F\| = \|f\| \), where \( \|F\| \) and \( \|f\| \) are the norms of \( F \) and \( f \) on \( X \) and \( S \), respectively. Which of the following is/are true?}

• A. \( F(x, y) = 2x + 3y \) is a Hahn–Banach extension of \( f \) to \( X \)
• B. \( F(x, y) = 2x + y \) is a Hahn–Banach extension of \( f \) to \( X \)
• C. \( f \) admits infinitely many Hahn–Banach extensions to \( X \)
• D. \( f \) admits exactly two distinct Hahn–Banach extensions to \( X \)

Hint:

For Hahn–Banach extensions, verify the norm-preserving condition and examine possible variations in extensions.

Answer 1

The Correct Option is B

Step 1: Verifying extensions. The functional \( F(x, y) = 2x + y \) is a Hahn–Banach extension because it satisfies \( F|_S = f \) and preserves the norm. However, \( F(x, y) = 2x + 3y \) does not satisfy the norm-preserving condition. Step 2: Multiple extensions. The Hahn–Banach theorem guarantees infinitely many extensions of \( f \) to \( X \), as extensions can vary in the \( y \)-component. Step 3: Conclusion. The correct answers are \( {(2), (3)} \).