Let C[0, 1] be the space of all continuous functions on [0, 1]. Recall that the uniform norm || · ||∞ and the L1-norm || · ||1 on C[0, 1] are defined by ||f||∞ = sup |f(t)|, ||f||1 = ∫ |f(t)|dt, f ∈ C[0, 1]. (1) Give an example of sequence {fn}n∈N ⊂ C[0, 1], such that ||fn||1 → 0, but ||fn||∞ ↛ 0. What about the implication ||fn||∞ → 0 ⇒ ||fn||1 → 0 for a sequence {fn}n∈N ⊂ C[0, 1]? Either prove this implication directly or construct a counterexample. (2) Prove that the formula ||f||0 := sup |tf(t)|, f ∈ C[0, 1] defines a norm on C[0, 1]. Give example of sequences {fn}n∈N and {gn}n∈N in C[0, 1], such that ||fn||1 → 0 but ||fn||0 ↛ 0, and ||gn||0 → 0 but ||gn||1 ↛ 0. (3) Prove that for an arbitrary vector space V equipped with norms || · || and || · ||' the following conditions are equivalent: (a) If {xn}n∈N ⊂ V is such that ||xn|| → 0, then ||xn||' → 0; (b) Every subset E that is bounded in the sense of || · || is bounded in the sense of || · ||'; (c) There exists a positive number α > 0, such that ||x||' ≤ α||x|| for any x ∈ V. (Recall that we say that a subset E of a normed space is bounded, if there exists a constant C > 0, such that E is contained in the ball BC = {x ∈ V : ||x|| ≤ C}.)