# Math Help - Hilbert Spaces: Prove Vo is a subspace

1. ## Hilbert Spaces: Prove Vo is a subspace

Let V be a vector space over C with a pre-inner product <v,w>
Let ||v|| be the semi-norm ||v|| = <v, v>.
(i) Show that Vo = {v element of V : ||v|| = 0} is a subspace of V .

2. Originally Posted by dazed
Let V be a vector space over C with a pre-inner product <v,w>
Let ||v|| be the semi-norm ||v|| = <v, v>.
(i) Show that Vo = {v element of V : ||v|| = 0} is a subspace of V .
Hint: The Cauchy–Schwarz inequality $|\langle v,w\rangle|\leqslant\|v\|\|w\|$ holds in a pre-inner-product space.