I have read all kinds of posts relative to Hilbert matrix such as Why does the inverse of the Hilbert matrix have integer entries?
Prove that a matrix is invertible
Why does the inverse of the Hilbert matrix have integer entries?
And recently I have encountered a problem relevant to Hilbert matrix.The original version is as follows: "Assume that $f\in C([0,1])$ and \begin{align*} \int_0^1f(x)d x=\int_0^1xf(x)dx=\cdots=\int_0^1x^{n-1}f(x)d x=1. \end{align*},Prove that $\int_0^1\big(f(x)\big)^2 x\geqslant n^2$"
Actually this problem is equal to proving the sum of all entries of the inverse of the Hilbert matrix with order $n$ is no less than $n^2$. And by observing the result that \begin{align*} H^{-1}_2=\begin{pmatrix} 4&-6\\ -6&12 \end{pmatrix},H^{-1}_3=\begin{pmatrix} 9&-36&30\\ -36&192&-180\\ 30&-180&180 \end{pmatrix} \end{align*} It's easy to see that the Hilbert matrix with order $n$($n=2,3$) is $n^2$.So I guess that It's true for all $n\in \mathbb{N}$.Can anyone give an elementary proof to my question?