# If a subspace is Banach and Quotient is Banach then the mother space is Banach.

Let $$X$$ be a Normed Linear Space and $$M$$ be a closed subspace of X. Assume that both $$M$$ and $$X/M$$ are banach spaces. Prove that $$X$$ is a banach space.

So firstly I assumed a Cauchy sequence $${(f_n)}$$ in $$X$$, then $${(f_n + M)}$$ is also cauchy in $$X/M$$. Suppose it converges to $$f+M$$. Then $$||(f_n-f)+M||$$ tends to 0 as n tends to infinity. So by the property of this norm there exists a sequence $$(g_n)$$ in $$M$$ such that $$(f_n+g_n)$$ converges to $$f$$. So I have to show that $$(g_n)$$ is cauchy and hence $$(f_n)$$ converges. But I'm facing problem in showing that $$(g_n)$$ is cauchy.

## 1 Answer

You know that $$f_n + g_n$$ is Cauchy in $$X$$ since it converges to $$f$$ in $$X$$ by the choice of $$g_n$$. By assumption, you also know that $$f_n$$ is a Cauchy sequence. It then follows that $$g_n = (f_n + g_n) - f_n$$ is a Cauchy sequence as it is a sum of two Cauchy sequences.

• Haha I'm a fool. Thank you btw. – MathCosmo May 10 at 14:47