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Banach Spaces

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Multiple Choice

Let $X$ be a Banach space and $T: X \rightarrow X$ be a linear operator. Which of the following statements is true?

  1. If $T$ is bounded, then it is continuous.
  2. If $T$ is continuous, then it is bounded.
  3. If $T$ is compact, then it is bounded.
  4. If $T$ is bounded, then it is compact.
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