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Let $mathcal{H}$ be a complex separable Hilbert space. We prove that if ${f_{n}}_{n=1}^{infty}$ is a Schauder basis of the Hilbert space $mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore, we investigate that ${z^{sigma^{-1}(n)}}_{n=1}^{infty}$ can never be a Schauder basis of $L^{2}(mathbb{T}, u)$, where $mathbb{T}$ is the unit circle, $ u$ is a finite positive discrete measure, and $sigma: mathbb{Z} rightarrow mathbb{N}$ is an arbitrary surjective and injective map.
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We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use $ell_{Psi}$-Hilbertian and $infty$-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition in a Hilbert space and introduce the class of spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and $M$-cotype of a Banach space and study the properties of unconditional Schauder decompositions in spaces possessing certain geometric structure.
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