1

The following is from Reproducing kernel Hilbert space section "Connection between RKHS with ReLU function".

We will work with the Hilbert space ${\displaystyle {\mathcal {H}}=L_{2}^{1}(0)[0,\infty )}$ of absolutely continuous functions with ${\displaystyle f(0)=0}$and square integrable (i.e. ${\displaystyle L_{2}}$) derivative.

I don't know the notation $L_{2}^{1}(0)[0,\infty )$ mean due to my little knowledge in math.

yllgl
  • 125
  • I'm not sure what the $1$ exponent is signifying, but it seems to be as they say, absolutely continuous functions on $[0,\infty)$ with $f(0)=0$ and square integrable derivative, so $\int_0^\infty f'(x)^2 dx < \infty$. – Ian Jun 01 '21 at 02:09
  • @Ian: It most likely means "one" derivative is involved, as in the usual Sobolev spaces. – Jose27 Jun 01 '21 at 02:20
  • So basically $\int_0^x f(y) dy$ for each $f \in L^2$. Makes sense. – Ian Jun 01 '21 at 03:04
  • @Ian:What does zero in $L_{2}^{1}(0)$ mean? Does it mean f(0)=0 ? – yllgl Jun 01 '21 at 07:56
  • Probably. ${}{}$ – Ian Jun 01 '21 at 08:32

0 Answers0