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

I'm asked to show that { $f:\mathbb R \rightarrow \mathbb R | f$ continuous} with operations

(a) $(f+g)(t)=f(t)+g(t)$

(b) $(rf)(t) = rf(t)$

make a vector space. I'm trying to show (i)-(iii) below, and I work through them, but I'm not sure if I'm understanding how they relate to (b) correctly.

(i) $r(v+w)=rv+rw$


(ii) $(r+s)v=rv+sv$


(iii) $(rs)v = r(sv)$


share|cite|improve this question
What do you denote by $\ast$, multiplication in $\mathbb{R}$? If so, why do you introduce that symbol? – k.stm Jan 22 '13 at 18:11
(a) and (b) + the set contains the zero vector is exactly how you show something is a space. Furthermore in your last line, you only need 1 scalar, not 2 – Bob Jan 22 '13 at 18:12
@K.Stm, yeah it's multiplication in the reals, I just introduced it o make it clearer, but it may have been a bad idea. – mlstudent Jan 22 '13 at 18:16
I just wanted to make sure that: The set in question is the space of continuous functions. The usual way to go about this is to check that the set of all functions $ℝ → ℝ$ is a vector space and the continuous functions form a subspace. – k.stm Jan 22 '13 at 18:33
Can you take the fact that $f+g$ and $\lambda f$ are continuous for granted or are you expected to prove it? – rschwieb Sep 11 '13 at 19:20

Your Answer


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

Browse other questions tagged or ask your own question.