Teaching JKPSC Lecturership Mock Test Series 2024-25 Mathematical Science Analysis Normed Linear Spaces
Let X be a real normed linear space. Let X0 = {x ∈ X : ∥x∥ = 1}. If X0 contains two distinct points x and y and the line segment joining them, then, which of the following statements is TRUE?
1
∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly independent
2
∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly dependent
3
∥x + y∥2 = ∥x∥2 + ∥y∥2 and x, y are linearly independent
4
∥x + y∥ = 2∥x∥∥y∥ and x, y are linearly dependent