We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons’s construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are $u$-ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.

Let $X$ be a Banach space. We give characterizations of when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when $ℱ(X,Y)$ is a $u$-ideal in $𝒲(X,Y)$ for every Banach space $Y$, when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X^{**})$ for every Banach space $Y$, and when $ℱ(Y,X)$ is a $u$-ideal in $𝒦(Y,X^{**})$ for every Banach space $Y$.

We investigate sequences and operators via the unconditionally p-summable sequences. We characterize the unconditionally p-null sequences in terms of a certain tensor product and then prove that, for every 1 ≤ p < ∞, a subset of a Banach space is relatively unconditionally p-compact if and only if it is contained in the closed convex hull of an unconditionally p-null sequence.

There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $in{f}_{ellipsoid\epsilon \subset B_E}{(vol\left(B_E\right)/vol\left(\epsilon \right))}^{1/n}\le c_{0}in{f}_{zonoidZ\subset B_F}{(vol\left(B_F\right)/vol\left(Z\right))}^{1/k}$ . The concept of volume ratio with respect to ${\ell }_p$-spaces is used to prove the following distance estimate for $2\le q\le p<\infty$: $su{p}_{F\subset {\ell }_p,dimF=n}in{f}_{G\subset L_q,dimG=n}d(F,G){\sim }_{c_{pq}}{n}^{(q/2)(1/q-1/p)}$.