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}