Part CCSIR NET December 2024the-two-norms-are-not-equivalent

The two norms are not equivalent

Consider the real vector space X = { $f : [0, 1] \to \mathbb{R}$ | f is continuous}, along with the norms ‖·‖ $_{1}$ and ‖·‖ $_{2}$ defined by ‖f‖ $_{1} = \int_{0}^{1}$ |f(x)|dx and ‖f‖ $_{2} = (\int_{0}^{1}$ |f(x)| $^{2}$ dx)^(1/2). For n ≥ 1 and x ∈ [0, 1], let $f_{n}(x) =$ nx $^{n}$ . Which of the following statements are true?

A.(‖ $f_{n}$ ‖ $_{1})$ is a convergent sequence.✓ Correct
B.(‖ $f_{n}$ ‖ $_{2})$ is a convergent sequence.
C.Both (‖ $f_{n}$ ‖ $_{1})$ and (‖ $f_{n}$ ‖ $_{2})$ are convergent sequences.
D.Neither (‖ $f_{n}$ ‖ $_{1})$ nor (‖ $f_{n}$ ‖ $_{2})$ is a convergent sequence.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests finite-dimensional intuition.

See pricing

50 are analysed free — try those first.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Related counterexample: L¹[0,1] ⊆ L²[0,1]

The two norms are not equivalent

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