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

Let $V$ be the space of real polynomials of degree $\leq n$.

a) Check the setting $(f(x),\,g(x))=\int_{0}^{1}f(x)g(x)\,dx$ turns $V$ to a Euclidean space.

b) If $n=1$, find the distance from $f(x)=1$ to the linear span $U=\langle x\rangle$.

share|cite|improve this question
What norm are you using? – Pratyush Sarkar Jan 11 '14 at 18:13
Thanks for your input. I have edited the question so that the full question is now included. – user112535 Jan 11 '14 at 18:16

Hint: Due to Pythagoras the square of the distance from $v$ to $\mathrm{span}(u)$ is $\|v\|^2-\langle\frac{u}{\|u\|},v\rangle^2$.

share|cite|improve this answer

Your Answer


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