# Sequence in a Hilbert space

Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in a Hilbert space $H$. Show that the following are equivalent:

1. the zero vector is the only vector orthogonal to all $x_n$, and

2. the subspace spanned by the $x_n$ is dense in $H$.

Thanks for the help!

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Let $M$ be the closure of the span of the $x_n$.
For $1 \Rightarrow 2$, think about the projection operator $P$ onto that closed linear subspace. If $M \neq H$, then there is a vector $h \notin M$. What do you know about $h - Ph$?
For $2 \Rightarrow 1$, think about $M^\perp$
You might need that for any $U\subseteq H$, the closure of span$(U)$ is just $U^{\perp\perp}$. –  Berci Sep 30 '12 at 12:39