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