Sachchidanand Prasad

12-05-2024

Problem: Let $V$ be an inner product space (need not be of finite dimension) and $W \subseteq V$ be a subspace. Let $W^\perp = \{ v \in V : \left\langle v,w \right\rangle = 0,\ \ \forall\ w \in W \} $.
  1. Show that $W \subseteq \left( W^\perp \right)^\perp .$
  2. Show that the equality need not hold in (A).
  3. Show that if $V$ is finite dimensional, then $\left( W^\perp \right)^\perp = W.$
linear-algebra inner-product-space
Solution: I encourage you to attempt to solve the problem today. The solution will be provided tomorrow. This will give you the opportunity to test your understanding of the problem and to improve your skills in solving similar problems in the future.