In this manuscript, we present several new results in finite and countable dimen- sional real Hilbert space 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 {fi}m i=1 in Rn are not dense in the family of m ≤ (2n − 2)-element sets of vectors in Rn for every finite n and the families of vectors which do norm retrieval in 2 are not dense in the infinite families of vectors in 2. We also show that if a Riesz basis does norm retrieval in 2 , then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.
