Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am just a beginner. So please be patient with me.

Consider $X=\{\frac 1 n : n \in \mathbb N\} \cup \{0\}$. Prove that the vector space of continuous functions on $X$ is linearly isomorphic to the space of convergent sequences in $\mathbb R$. Using this result, conclude that the set of convergent sequences in the NLS of all bounded real sequences under the sup norm is complete.

share|cite|improve this question
up vote 3 down vote accepted

$\bf Hint:$ If $f: X\to\mathbb R$ is continuous then $\lim_{n\to\infty}f(\frac 1 n)=f(0)$ hence $f$ is a convergent sequence. On the other hand, if $(a_n)$ converges to $a$ then the function $f: X\to\mathbb R$ given by $\frac 1 n\to a_n$ and $0\to a$ it is easily seen to be a continuous function.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.