  In this manuscript, we present several new results in finite and countable
dimensional real Hilbert space phase retrieval and norm retrieval by vectors
and projections. We make a detailed study of when hyperplanes do norm
retrieval. Also, we show that the families of norm retrievable frames
$\{f_{i}\}_{i=1}^{m}$ in $\mathbb{R}^n$ are not dense in the family of $m\leq
(2n-2)$-element sets of vectors in $\mathbb{R}^n$ for every finite $n$ and the
families of vectors which do norm retrieval in $\ell^2$ are not dense in the
infinite families of vectors in $\ell^2$. We also show that if a Riesz basis
does norm retrieval in $\ell^2$, then it is an orthogonal sequence. We provide
numerous examples to show that our results are best possible.

Functional Analysis