The Blaschke–Kakutani result characterizes inner product spaces , among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace there is a norm 1 linear projection onto . In this paper, we determine which closed neighborhoods of zero in a real locally convex space of dimension at least 3 have the property that for every 2 dimensional subspace there is a continuous linear projection onto with .
An investigation is carried out of the compact convex sets X in an infinite-dimensional separable Hilbert space , for which the metric antiprojection from e to X has fixed cardinality n+1 ( arbitrary) for every e in a dense subset of . A similar study is performed in the case of the metric projection from e to X where X is a compact subset of .
We give a simple proof of the relation between the spectra of the difference and product of any two idempotents in a Banach algebra. We also give the relation between the spectra of their sum and product.
Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has length (1 - ε)n. On the other hand, there is a frame which does not contain bases with brackets.
We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution...
Let be a self-affine measure associated with an expanding matrix M and a finite digit set D. We study the spectrality of when |det(M)| = |D| = p is a prime. We obtain several new sufficient conditions on M and D for to be a spectral measure with lattice spectrum. As an application, we present some properties of the digit sets of integral self-affine tiles, which are connected with a conjecture of Lagarias and Wang.