Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

$X,Y$ be complex norm linear space which are not necessarily complete, let $T:X\rightarrow Y$ be a linear map such that $\{T(x_n)\}$ is Cauchy in $Y$ whenever $\{x_n\}$ is Cauchy in $X$, we need to show $T$ is continuous.

My Attempt: construct two new sequence $y_n,z_n$ such that $$y_1=T(x_1)$$ $$z_1=T(x_2)$$ $$y_2=T(x_3)$$ $$z_2=T(x_4)$$ $$\dots$$ $$y_n=T(x_{2n-1})$$ $$z_n=T(x_{2n})$$ let $x_n\rightarrow c\in X$ we need to show $T(x_n)\rightarrow T(c)\in Y$, am I going in right path?if not please help!

share|cite|improve this question

For given $x\in X$, let $\{ x_n\}$ be a sequence convergent to $x$. Define $y_n$ by setting $y_n=x_{n/2}$ when $n$ is even and $y_n=x$ otherwise. Again we have $y_n\rightarrow x$ and also $\{y_n\}$ is Cauchy. Therefore $T(y_n)$ is Cauchy which easily implies that $T(x_n)\rightarrow T(x)$.

share|cite|improve this answer
could you please tell me was my attempt wrong? can not be proceeded in my way? – Un Chien Andalou Feb 10 '13 at 14:03
You define two Cauchy sequence from one and I do not see how it helps to prove they are convergent to some point. – Vahid Shirbisheh Feb 10 '13 at 14:40

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.