Let B be a real Banach space and B* its dual. Let the norms on B and B* be denoted by /) .I] and I/ . //*, respectively. If B is a smooth Banach space then the map u: B--f B*, defined by U(X) = ]I x 11 G3c,,,za if x # 0, and o(O) = 0, G, denoting the GIteaux gradient of the norm at x, is known as the extended spherical image map. It is known that the extended spherical image map is a homeomorphism if and only if jj . /I and 11 . I/* are FrCchet differentiable away from the origin [2]. It is also known that if the norms of B and B* are twice FrCchet differentiable away from the origin (in particular if the extended spherical image map is a Cl diffeomorphism) then B is isomorphic to a Hilbert space [5, Theorem 53. The purpose of the present note is to obtain a complete characterization of inner-product spaces in terms of smoothness and to show that the theorem in [5] cited above is the best possible in the sense that the isomorphism cannot be replaced by an isometric isomorphism. This also solves the following problem posed by Phelps in the affirmative: Are there Banach spaces B not isometric with inner-product spaces such that the norms of B and B* are twice FrCchet differentiable away from the origin? Before proceeding to the main results of the paper, a few definitions and known facts are stated. B denotes a real Banach space and B *, B** are the first and second duals of B. g(B) A?(B, B*) denote, respectively, the Banach space of bounded bilinear functionals on B and the Banach space of bounded linear operators on B into B* with the usual supremum norms. The norms of B and B* are denoted respectively by Ij . II and 11 . II*, while the norms of &Y(B) and 3(B, B*) will be denoted by II . 11 itself as there will be no occasion for confusion. Since the mapping m: 9(B, B*) --f W(B) defined by m(T) (x, y) = T(x) (y) for x, y E B is a linear isometry we identify these spaces. U( U*), S(S*) denote, respectively, the unit ball and unit sphere of B(B*).