Sharp Hardy's Inequalities in Hilbert Spaces

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy’s inequalities $${\int }_a^b{(\frac{1}{x}{\int }_a^{x}f(t)dt)}^{2}dx\le d(a,b){\int }_a^b[f(x){]}^{2}dx$$ and $$\sum _{k=1}^n(\frac{1}{k}\sum _{j=1}^k{a}_j{)}^2\le d_n\sum _{k=1}^n{a}_k^2\mathrm{.}$$ The exact constant $d(a,b)$ and the precise rate of convergence of $d_n$ are established and the extremal function and the ``almost extremal’’ sequence are found.