Quiz on Normed Linear Spaces

Which of the following is a normed linear space?

• ℝ with absolute value as norm

• ℝ² with Euclidean norm

• ℂⁿ with p-norm (1 ≤ p ≤ ∞)

• All of the above

A norm ‖·‖ on a vector space V satisfies:

• Triangle inequality

• Homogeneity

• Positive definiteness

• All of the above

If ‖x‖ = 0 implies x = 0, then the norm is said to be:

• Linear

• Positive definite

• Continuous

• Homogeneous

The norm ‖x‖ = max{|x₁|, |x₂|, ..., |xₙ|} on ℝⁿ is called:

• Euclidean norm

• ℓ¹ norm

• Sup norm (∞-norm)

• Hilbert norm

A normed space is a metric space with metric defined by:

• d(x, y) = x + y

• d(x, y) = ‖x - y‖

• d(x, y) = ‖x + y‖

• d(x, y) = |x - y|²

A normed linear space is called complete if:

• Every Cauchy sequence converges

• Every bounded sequence converges

• Every sequence converges

• Every function is continuous

A complete normed linear space is called:

• Metric space

• Inner product space

• Banach space

• Hilbert space

The space ℓ² of square-summable sequences is:

• A Banach space

• Not normed

• A Hilbert space

• Both A and C

Which of the following is NOT a norm?

• ‖x‖ = |x| on ℝ

• ‖x‖ = √(x₁² + x₂²) on ℝ²

• ‖x‖ = x₁ + x₂ on ℝ²

• ‖x‖ = max(|x₁|, |x₂|)

In a normed linear space, the norm is always:

• Differentiable

• Convex

• Constant

• Subadditive only