Characterization of Real Inner Product Spaces by Means of Symmetrical Bilinear Forms

We investigate those normed spaces E for which the norm of any continuous symmetric bilinear form on E has the same value as the norm of the associated homogeneous polynomial. In the real case we show that this property characterizes an inner product space (i.p.s). We also give some other characterizations of an i.p.s., through symmetric bilinear forms.