Given a sequence of i.i.d. random variables {X,X n ;n≥1} taking values in a separable Banach space (B, ∥⋅∥) with topological dual B ∗, let X n ( r ) = X m $X_{n}^{(r)}=X_{m}$ if ∥X m ∥ is the r-th maximum of {∥X k ∥;1≤k≤n} and ( r ) S n = S n − ( X n ( 1 ) + ⋯ + X n ( r ) ) ${}^{(r)}S_{n}=S_{n}-(X_{n}^{(1)}+\cdots +X_{n}^{(r)})$ be the trimmed sums when extreme terms are excluded, where S n = ∑ k = 1 n X k . In this paper, it is stated that under some suitable conditions, lim n → ∞ 1 2 log log n max 1 ≤ k ≤ n ∥ ( r ) S k ∥ k = σ ( X ) a . s . , $\lim _{n\to \infty }\frac {1}{\sqrt {2\log \log n}}\max _{1\le k\le n}\frac {\|^{(r)}S_{k}\|}{\sqrt {k}}=\sigma (X)~~~\mathrm {a.s.},$ where σ 2 ( X ) = sup f ∈ B 1 ∗ E f 2 ( X ) $\sigma ^{2}(X)=\sup _{f\in B_{1}^{*}}\text {\textsf {E}} f^{2}(X)$ and B 1 ∗ is the unit ball of B ∗.