Inner product space functional analysis

show that y is orthagonal to x_n and x_n---->x together imply x is othagonal to y

Note: an element x of an inner product space X is said to be orthagonal to an element y belonging to X if =0 similarly for subsets A of X we write x is orthagonal to A if for all a belonging to A we have =0

To show that =0, it suffices to show that || < e for all positive numbers e. Fix any e>0. Since x_n --> x, we can choose N large enough that |x - x_N| < e/|y|. Then

|| = | + | <= || + || = ||,

since x_n is orthogonal to y. By the Cauchy-Schwartz inequality,

|| <= || <= |x - x_N| * |y| < e,

which completes the proof.