𝑥∈𝑋and therefore𝑓+𝑔∈𝐵(𝑋). Hence,𝐵(𝑋) is closed under addition andscalar multiplication; it is a subspace of the linear space𝐹(𝑋). We conclude that𝐵(𝑋) is a normed linear space.3. To show that𝐵(𝑋) is complete, assume that (𝑓𝑛) is a Cauchy sequence in𝐵(𝑋).For every𝑥∈𝑋∣𝑓𝑛(𝑥)−𝑓𝑚(𝑥)∣ ≤ ∥𝑓𝑛−𝑓𝑚∥ →0Therefore, for𝑥∈𝑋,𝑓𝑛(𝑥) is a Cauchy sequence of real numbers. Sinceℜiscomplete, this sequence converges. Define the function𝑓(𝑥) = lim𝑛→∞𝑓𝑛(𝑥)We need to show∙ ∥𝑓𝑛−𝑓∥ →0 and∙𝑓∈𝐵(𝑋)(𝑓𝑛) is a Cauchy sequence. For given𝜖 >0, choose𝑁such that∥𝑓𝑛−𝑓𝑚∥< 𝜖/2for very𝑚, 𝑛≥𝑁. For any𝑥∈𝑋and𝑛≥𝑁,∣𝑓𝑛(𝑥)−𝑓(𝑥)∣ ≤ ∣𝑓𝑛(𝑥)−𝑓𝑚(𝑥)∣+∣𝑓𝑚(𝑥)−𝑓(𝑥)∣≤ ∥𝑓𝑛
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