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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!

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

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