A norm \(\lVert \cdot\rVert\) induces an inner product via the polarization identity if and only if it satisfies the parallelogram law .

Proof
  • \( \Rightarrow \): Clear.
  • \( \Leftarrow \): Define \( \langle \cdot, \cdot \rangle \) via the polarization identity and prove the properties of an inner product [1].

References Link to heading

  1. D. Werner, Funktionalanalysis. Berlin, Heidelberg: Springer Berlin Heidelberg, 2018. doi:10.1007/978-3-662-55407-4