Find a vector $\displaystyle (w_1,w_2,w_3,...)$ that is orthogonal to $\displaystyle v=(1,\frac{1}{2},\frac{1}{4},...)$. Compute its length $\displaystyle ||w||$.
There are infinite Hilbert spaces ( to add "with inner product" is futile: ANY Hilbert space is a vector space with an inner product). I can think of at least
two different Hilbert spaces with their inner products to which your vector belongs, so again: what vector space with what iiner product?
Tonio
That's what I also thought, but then you already know what to look for......
Idea: $\displaystyle \sum\limits^\infty_{n=0}\frac{1}{4^n}=\frac{4}{3} \, , \, \, \,\sum\limits^\infty_{n=1}\frac{1}{2^{2n-1}}=2\sum\limits^\infty_{n=1}\frac{1}{4^n}=2\cdot \frac{1}{3}=\frac{2}{3}$
Tonio