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 Mar 17 awarded Popular Question May 12 comment Continuous representation restricts to homomorphism $G \to O(n)$ Do you remember how this was done for G finite? Can we do a similar thing...? May 4 comment How is the Fourier transform “linear”? Indeed your standard linear functions of the form $f(x) = ax + b$ aren't actually linear at all when $b \ne 0$ (since linear functions map 0 to 0). These straight line functions comprise the more general class of affine transformations Apr 19 comment Integral metric. The space C[0,1] equipped with the above norm isn't a Hilbert space: it fails on both counts. It isn't complete, consider $f_m(x) = 0$ for $0 \le x \le \frac{1}{2} - m$ and $f_m(x) = 1$ for $\frac{1}{2} + m \le x \le 1$ and f being a straight line segment joining 0 and 1 between these two intervals. It's a Cauchy sequence, but it would necessarily converge to a function with a discontinuity at x= 1/2. So we could consider its completion, but this space cannot be an inner product space since it does not obey the en.wikipedia.org/wiki/Parallelogram_law Apr 19 answered Integral metric. Apr 15 comment How to prove that the inverse exists on the whole space X? A linear map is injective if and only if it has trivial kernel. So suppose there is some non-zero x with (I-T)x = 0. Then x = Tx so $||Tx||/||x|| \ge 1 \Rightarrow ||T|| \ge 1$, a contradiction. Apr 13 answered Evaluation of $\sum_{x=1}^{\infty}x^{-x}$ Apr 12 awarded Supporter Apr 12 awarded Student Apr 12 asked Cutesy Applications of Fermat's Last Theorem (or others) Apr 12 awarded Teacher Apr 12 answered Mapping a variable having very vast range to the interval (0,1)