a concrete example of something that's clearly not a valid kernel would be if you find an input x such that k of x, x " and this is minus one, for example " then this is an example of something that's clearly not a valid kernel, because minus one cannot possibly be equal to phi of x transpose phi of x, and so this would be one of many examples of functions that will fail to meet the conditions of this theorem, because inner products of a vector itself are always greater than zero.