Abstract. It is shown that the complex tangent space $R_a$ at a point $a$ on the surface of the unit ball $A_1$ in a complex Banach space $A$ coincides with the complex linear span $$\mathop{\rm lin}_{{\msbm C}}(\{{\rm i} a\}^{\rlap{\sqcap}\sqcup} \cap\{a\}^{\rlap{\sqcap}\sqcup} \cap A_1)$$ of the set $\{{\rm i} a\}^{\rlap{\sqcap}\sqcup} \cap\{a\}^{\rlap{\sqcap}\sqcup} \cap A_1$, where, for a subset $L$ of $A$, \[L^{\rlap{\sqcap}\sqcup} = \{a \in A: \|a \pm b\| = \max\{\|a\|,\|b\|\}, \forall b \in L\}\] is the M-orthogonal complement of $L$. It is also shown that if $B$ is a holomorphically rigid closed subspace of $A$ then $B^{\rlap{\sqcap}\sqcup}$ is equal to $\{0\}$. In the special case in which $A$ is a JBW$^*$-triple and $B$ is a weak$^*$-closed subtriple of $A$, it is shown that the M-orthogonal complement $B^{\rlap{\sqcap}\sqcup}$ of $B$ coincides with the algebraic annihilator $B^{\perp}$ of $B$, that the complex tangent space $R_{L_B}(A)$ at the set $L_B$ of elements of $B$ of unit norm is weak$^*$-closed and also coincides with $B^{\rlap{\sqcap}\sqcup}$, that a second tangent space $T^n_{L_B}(A)$ at $L_B$ is weak*-closed and coincides with the algebraic kernel $\mathop{\rm Ker}(B)$ of $B$, and that $B$ is holomorphically rigid in $A$ if and only if $B^{\rlap{\sqcap}\sqcup}$ is equal to $\{0\}$.

AMS Subject Classification (1991): 46L70; 17C65, 46G20

Keyword(s): M-structure, tangent spaces, holomorphic rigidity

Received April 8, 2003, and in revised form October 14, 2003. (Registered under 5811/2009.)