Is this extension a Hilbert space?

Question

Let $V$ be an inner product space over $\mathbb{F}$.

Let $H$ be a complete subspace of of $V$ and $x\in V\setminus H$

Define $K= span(H\cup \{x\})$.

Is $K$ a Hilbert space? How do I prove it?

Answer 1

Let $x = x_1 + x_2$ with $x_1 \in H$ and $x_2 \in H^\bot$ (why do these exist?).

Then you can assume w.l.o.g. that $x= x_2$ (why?).

From there you should be able to show that if $(h_n + \alpha_n x)_n$ is Cauchy, then $(h_n)_n$ and $(\alpha_n)_n$ are both Cauchy (at least if $x \neq 0$, but this follows from $x\notin H$), which should help you.