The Banach spaces $L(X,Y),K(X,Y),{{L}_{{{w}^{*}}}}({{X}^{*}},Y)$, and ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ are studied to determine when they contain the classical Banach spaces ${{c}_{o}}$ or ${{l}_{\infty }}$. The complementation of the Banach space $K(X,Y)$ in $L(X,Y)$ is discussed as well as what impact this complementation has on the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in $K(X,Y)$ or $L(X,Y)$. Results of Kalton, Feder, and Emmanuele concerning the complementation of $K(X,Y)$ in $L(X,Y)$ are generalized. Results concerning the complementation of the Banach space ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ in ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$ are also explored as well as how that complementation affects the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ or ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$. The ${{l}_{p}}$ spaces for $1\,=\,p\,<\,\infty $ are studied to determine when the space of compact operators from one ${{l}_{p}}$ space to another contains ${{c}_{o}}$. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.