I am working through an example in "A Course on Functional Analysis" by John Conway and I am having trouble justifying a certain fact stated in an example:
Let $X$ be the collection of all sequences $\{a_n : n\geq 1\}$ of scalars from $\mathbb{F}$ such that $a_n = 0$ for all but a finite number of values $n$.
In this text we consider $\mathbb{F}$ as either the real numbers or complex numbers, but not any other field. The example then gives the following inner products on $X$.
$$ \langle a_n,b_n\rangle = \sum_{n=1}^\infty a_n\overline{b_n} $$ $$ \langle a_n, b_n\rangle = \sum_{n=1}^\infty \frac1n a_n \overline{b_n} $$ $$ \langle a_n, b_n\rangle = \sum_{n=1}^\infty n^5 a_n \overline{b_n} $$
The example proclaims that these define all the inner products on $X$. This brought two questions to my attention.
- How does one find all of the inner products on a vector space?
- Why are these the only inner products on $X$ for the example?
First, question 1. There are 6 properties that a function $\langle\cdot, \cdot\rangle:X \times X \rightarrow \mathbb{F}$ has to satisfy in order to be an inner product:
$$ \langle\alpha x + \beta y, z\rangle = \alpha \langle x,z\rangle + \beta \langle y,z\rangle $$ $$ \langle x, \alpha y + \beta z\rangle = \overline{\alpha}\langle x,y\rangle + \overline{\beta}\langle x,z\rangle $$ $$ \langle x,x\rangle \geq 0 $$ $$ \langle x,y\rangle = \overline{\langle y,x\rangle} $$ $$ \langle x,0\rangle = \langle 0,y\rangle = 0 $$ $$ \langle x,x\rangle = 0 \implies x = 0 $$
However, just these properties alone doesn't provide many hints into how to determine the functions. Is there a good strategy for finding the inner products of a vector space?
Secondly, question 2. I don't see why these functions in particular form the collection of all inner products on $X$. At first glance it appears as though any function $f:\mathbb{N} \rightarrow \mathbb{F}$ that is non-zero for all $n\in \mathbb{N}$ would satisfy the conditions for this example. To show this if we let $\langle a_n,b_n\rangle = \sum_{n=1}^\infty f(n)a_n\overline{b_n}$ we have:
$$ \langle\alpha a_n + \beta b_n, c_n\rangle = \sum_{n=1}^{\infty} f(n)(\alpha a_n + \beta b_n)c_n = \alpha \sum_{n=1}^\infty f(n)a_n c_n + \beta \sum_{n=1}^\infty f(n)b_nc_n $$
As required. The second condition is similarly satisfied. The third can be seen since we chose $f(n) \neq 0$ for all $n\in \mathbb{N}$, the fourth and fifth follows easily as well. Lastly the sixth condition is satisfied since we range $a_n$ and $b_n$ over all the natural numbers, and $f(n)$ is non-zero. From this I would have to conclude that there are infinitely many inner products on $X$. What am I missing?
Edit: it turns out I misread the passage the examples above "all define inner products on $X$", not "define all inner products." It makes much more sense now!
Let $V_k = \{ x:\mathbb N \to \mathbb F \mid \forall n > k : x_n = 0 \}$ and $V = \bigcup_k V_k$. Then, any inner product $\beta$ on $V$ is also an inner product on $V_k$. But inner products on finite dimensional spaces are just positive definite matrices.