Suppose $A$ and $B$ are orthogonal Hilbert subspaces. To find the orthogonal complement of the direct sum with projections, is the following true?

\begin{equation} (A \bigoplus B)^\perp = \prod(A^\perp | B^\perp), \end{equation} where $\prod(A^\perp | B^\perp)$ is the projection of $A^\perp$ onto $B^\perp$.

  • $\begingroup$ What is this notation on the right hand side? Anyway it should simply be $A^\perp\cap B^\perp$. $\endgroup$
    – Berci
    Commented May 20, 2022 at 20:18
  • $\begingroup$ I have made the question hopefully clearer. $\endgroup$
    – jsmath
    Commented May 20, 2022 at 20:50
  • 1
    $\begingroup$ No, not any cleaner. I would use e.g. $p_A$ and $p_B$ for the projections and both have the whole ambient space as domains. $\endgroup$
    – Berci
    Commented May 20, 2022 at 21:02


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