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GATE 2022 MA – Question 15

Functional Analysis · Normed and Banach Spaces · 1 mark · Multiple choice

The unit sphere of a real normed space contains distinct x,y and their entire joining segment. Which holds?

  1. ‖x+y‖=‖x‖+‖y‖ and x,y are independent.
  2. ‖x+y‖=‖x‖+‖y‖ and x,y are dependent.
  3. ‖x+y‖²=‖x‖²+‖y‖² and x,y are independent.
  4. ‖x+y‖=2‖x‖‖y‖ and x,y are dependent.

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Correct answer: (A) ‖x+y‖=‖x‖+‖y‖ and x,y are independent.

Explanation

The midpoint has norm 1, so ‖x+y‖=2=‖x‖+‖y‖. If x,y were dependent unit vectors over ℝ, they would be equal or opposite. Equality is excluded, and opposition would put zero in the segment, outside the unit sphere. They are independent. **A**.

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