# Closed subspace of a Normed Space

Let $$W$$ be a closed subspace of a Normed Space $$X$$. Let $$x\in X$$and $$x\notin W$$ , then $$Span[{x}]\oplus W$$ is a closed ?

Any proof? Thanks!

• What did you try? – José Carlos Santos Oct 11 '18 at 11:12
• Any efforts?${}$ – Saad Oct 11 '18 at 11:12
• I was trying using sequences but I can not prove it at all – Studentmathever Oct 11 '18 at 11:15

Hints: suppose $$a_nx+w_n \to z$$ where $$w_n \in W$$ for all $$n$$. If $$|a_n| \to \infty$$ then $$x+\frac 1 {a_n} w_n \to 0$$ so $$\frac 1 {a_n} w_n \to -x$$ which gives the contradiction that $$x \in W$$. In the contrary case some subsequence of $$a_n$$ converges to a finite limit. Now try to complete the proof.