2. (a) Let W be a vector space with inner product (,) and the norm ||x|| = √(x,x). Prove that, for u, w € W, ||u+w||² + ||uw||² = 2||u||² + 2||w||². Let V be a vector space. A norm is a function v→ ||v|| that satisfies |v|| ≥ 0, for v € V. ||v|| = 0 if and only if v = 0. ||rv|| = |r|||v||, for v EV and r ER. P4: ||u+v|≤|u|| + ||v||, for v € V. (b) Definition: P1: P2: P3: -IL- I₂ The 1-norm and the infinity norm on R² are defined as For example, for BI = √+ is called the 2-norm on R². = |2₁| + |₂|| and AL I2 i. Show that the 1-norm satisfies properties P1 to P4. ii. Show that the 1-norm does not satisfy Equation (1). = = max(₁, ₂), respectively.

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