Given x,y elements of a hilbert space H, how do we conclude that x = y? Since there is an inner product, we can say that x = y only if (x,z) = (y,z) for all z in H. But is there a definition of equality which does not depend on the inner product? A hilbert space is a special instance of vector space. According to wikipedia https://en.wikipedia.org/wiki/Vector_space#Definition the definition of a vector space is given in terms of properties relating elements of the space that depend on a prior understanding of equality, e.g., commutativity with respect to addition. What does equals mean in this case?