GATE 2026 MA – Question 19
X = C¹([0,1],ℝ). Define $p_1(x)=\sup|x(t)|$, $p_2(x)=\sup|x^\prime(t)|$, and p₃ = p₁+p₂. Which statement is true?
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Correct answer: (D) (X,p₃) is Banach.
Explanation
A p₃-Cauchy sequence has uniform limits x_n→f and x_n′→g, both continuous. Passing to limits in $x_n(t)=x_n(0)+\int_0^t x_n^\prime(s)ds$ gives f′ = g, so f ∈ C¹ and convergence holds in p₃. Hence **D**. The derivative-only p₂ is a seminorm, not a norm on X, since nonzero constants have p₂ = 0; the source’s blanket “norms” description is inaccurate. Uniform convergence alone need not preserve differentiability, so A fails.