What common definition of norm on the space of analytic functions makes the basis $e_n=\frac{x^n}{n!}$ orthonormal? I mean, this basis, with factorials is very useful as it is the basis of Taylor expansion. But it is not orthonormal under usual definitions of norm ($\int_a^b \sqrt{f(x)^2}dx$). I wonder, how it should be modified so to make the basis normal.
You can extend it to complex domain and then u will have an upper traingular form for the matrix: $a_{ij}=\langle z^i,z^j\rangle $ since for $i&gt;j$ $a_{ij}=\langle  z^i,z^j\rangle =0$ by cauchy integral theorem where the inner product is defined as $\langle z^i,z^j\rangle =\int_C z^{i-j}\,\mathrm{d}z$ for a closed curve $C$.
If u define the same inner product for a closed curve not including origin, it might work as diagonal matrix also.
You can convert ur intution into equation. Another definition of inner product for real analytic functions only would be: would be $&lt;f,g&gt;=\sum_i D^i(f)|_{x=a} D^i(g)|_{x=a}$ where $D$ is the derivative operator. In this inner product $(x-a)^i$ and $(x-a)^j$ are orthogonal.
You can extend it to complex domain and then u will have an upper traingular form for the matrix: $a_{ij}=\langle z^i,z^j\rangle $ since for $i&gt;j$ $a_{ij}=\langle  z^i,z^j\rangle =0$ by cauchy integral theorem where the inner product is defined as $\langle z^i,z^j\rangle =\int_C z^{i-j}\,\mathrm{d}z$ for a closed curve $C$.
If u define the same inner product for a closed curve not including origin, it might work as diagonal matrix also.
You can convert ur intution into equation. Another definition of inner product for real analytic functions only would be: would be $&lt;f,g&gt;=\sum_i D^i(f)|_{x=a} D^i(g)|_{x=a}$ where $D$ is the derivative operator. In this inner product $(x-a)^i$ and $(x-a)^j$ are orthogonal.