The proof goes as follows:(adsbygoogle = window.adsbygoogle || []).push({});

For contradiction, assume there exists |s|< ∞ such that s = {e_{1}, e_{2}, ... , e_{n}} and span(s} = ℝ^∞.

The above makes at least some sense to me. The proof goes on...

Let m > n andu= e_{n+1}+ e_{n+2}+ ... + e_{m}

u[itex]\notin[/itex] span(s),u[itex]\in[/itex] s

Because {e_{1}, e_{2}, ... , e_{n}} [itex]\notin[/itex] s, this implies a contradiction. Therefore ℝ^∞ is infinite-dimensional.

I don't see what we just did, and I don't see how what we just did proves that R^infinity is infinite-dimensional. Please help!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Help me understand this proof of R^infinity's infinite-dimensionality

**Physics Forums | Science Articles, Homework Help, Discussion**