[SOLVED] A formulation of continuity for bilinear forms 1. The problem statement, all variables and given/known data My HW assignment read "Let H be a real Hilbert space and a: H x H-->R be a coninuous coersive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M>0 such that |a(x,y)|<M||x|| ||y|| (iii) there exists B such tthat a(x,x)>a||x||^2" So apparently, condition (ii) is the statement about continuity. But I fail to see how this statement is equivalent to "a is continuous". I see how (ii) here implies continuous, but not the opposite. 3. The attempt at a solution Let z_n = (x_n,y_n)-->0. Then x_n-->0 and y_n-->0. So |a(x,y)|<M||x|| ||y|| implies a(x,y)-->0. a is thus continuous at 0, so it is so everywhere, being linear.