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- W4366449228 abstract "Abstract Kuelbs cite{K} has shown that every infinite-dimensional separable Banach space $mathcal{B}$ can always be densely and continuously embedded in a separable Hilbert space $mathcal{H}$. If $mathcal{L}[mathcal{B}]$ is the set bounded linear operators on $mathcal{B}$, in this paper we prove the following: begin{enumerate}item For each $u in mathcal{B}$, there exists a semi-inner product $[, cdot, u]_z$ generated by bounded linear functional $u_z^* in mathcal{B}^*$ and a constant $c_u$ such that $u_z^*(v)=[v,u]_z= c_u(v , u)_mathcal{H}$ for every $v in mathcal{B}$.item If $mathbb{A} ne mathcal{B}$ is closed, there exists $mathbb{A}^bot subset mathcal{B}$ disjoint, and $mathcal{B} = mathbb{A} oplus mathbb{A}^ bot$.item For $A in mcL[mathcal{B}]$ there exists $A^* in mcL[mathcal{B}]$ and $(A^*A)^*=A^*A$.item $mathcal{L}[mathcal{B}] subset mathcal{L}[mathcal{H}]$ as a continuous dense embedding.item $mathcal{B} times mathcal{B}^*$ has an Auerbach basis.item Every compact operator on $mathcal{B}$ is the limit of operators of finite rank without assuming that $mathcal{B}$ has a Schauder basis.item The Schatten class $mathbb{S}_p [mathcal{B}]$ exists for each $p in [1, iy]$, $mathbb{S}_p [mathcal{B}] subset mathbb{S}_p [mathcal{H}]$ as a continuous dense embedding, $bar{A} in mathbb{S}_p [mathcal{H}]$ if and only if its restriction $A in mathbb{S}_p [mathcal{B}]$, and ${left| {bar A} right|_p^mathcal{H}} = {left| A right|_p^mathcal{B}}$. end{enumerate}" @default.
- W4366449228 created "2023-04-22" @default.
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- W4366449228 date "2023-04-19" @default.
- W4366449228 modified "2023-10-16" @default.
- W4366449228 title "Almost a Hilbert Space" @default.
- W4366449228 doi "https://doi.org/10.21203/rs.3.rs-2829023/v1" @default.
- W4366449228 hasPublicationYear "2023" @default.
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