$\ell_2$ is an infinite-dimensional inner product space consisting of all infinite sequences of the form $(x_1,x_2,\ldots) $ satisfying that $\sum_{n=1}^\infty x_n^2 = x_1^2+x_2^2+\cdots$ converges. Given $x=(x_1,x_2,\ldots)$ and $y=(y_1,y_2,\ldots)$ in $\ell_2$, $\ell_2$ has the inner product: $\langle x,y \rangle = \sum_{n=1}^\infty x_ny_n = x_1y_1+x_2y_2+\cdots$ which is also an infinite series.
Given the two sequences in $\ell_2: x=(1,\frac23,\frac49,\frac8{27},\ldots) $ and $ y (1,\frac34,\frac49,\frac{27}{64},...)$
Find $\langle x,y \rangle$, $\|x\|$,$\|y\|$
Let $S$ be the subspace of $\ell_2$ spanned by $x$. Find $\operatorname{proj}_s y$ and $\operatorname{proj}_{s^\perp}y$ (as in the projection of $y$ onto $S$ and the projection of $y$ onto $S^\perp$
Doing Part 1 is fairly easy, I got that $\langle x,y \rangle=\sum_{n=1}^\infty (\frac12)^{n-1}$ which is just 2, $\|x\|^2=\langle x,x\rangle=\frac95$,and $\|y\|^2=\langle y,y\rangle =\frac{16}7$, but I have no clue how to even start Part 2. Any help would be appreciated
As in finite dimension geometry you have: $$\operatorname{proj}_S y = \langle y,x \rangle \frac{x}{\Vert x \Vert^2}$$ and: $$\operatorname{proj}_{S^\perp} y = y-\operatorname{proj}_{S} y$$