Exercise 11.1.6

. In this exercise, we will work out the details of Example 11.7. We must show that $\mathbf{Hilb}_{\mathbb{R}}$ is a vector space. We will do this by showing that it is a subspace of $\mathbf{Seq}_{\mathbb{R}}$. Further, we must show that the inner product is well-defined. This requires some knowledge of convergent series from calculus.

1. Assume $a=(a_{0},a_{1},\ldots)$ and $b=(b_{0},b_{1},\ldots)$ are square summable sequences. Show that the series $a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}+\ldots$ converges to a real number. Hint: consider the series $\left|a_{0}b_{0}\right|+\left|a_{1}b_{2}\right|+\ldots$ and use the Cauchy-Schwarz inequality and the absolute convergence test.

2. Using the result of part (a), show that $\mathbf{Hilb}_{\mathbb{R}}$ is a subspace of $\mathbf{Seq}_{\mathbb{R}}$.

3. Show that the operation $\langle a,b\rangle=a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}+\ldots$ is an inner product on $\mathbf{Hilb}_{\mathbb{R}}$.