Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in a Hilbert space $H$. Show that the following are equivalent:
the zero vector is the only vector orthogonal to all $x_n$, and
the subspace spanned by the $x_n$ is dense in $H$.
Thanks for the help!
Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in a Hilbert space $H$. Show that the following are equivalent:
the zero vector is the only vector orthogonal to all $x_n$, and
the subspace spanned by the $x_n$ is dense in $H$.
Thanks for the help!
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 $M^\perp$