Do ${\mathbb R}^\infty$, ${\mathbb Q}^\infty$, and ${\mathbb C}^\infty$ admit dot/inner products?

Question

I wonder if the vector spaces ${\mathbb R}^\infty$, ${\mathbb Q}^\infty$, and ${\mathbb C}^\infty$ admit dot/inner products. Maybe even extensions of the usual dot/inner products from ${\mathbb R}^n$, ${\mathbb Q}^n$, and ${\mathbb C}^n$?

Answer 1

If by $\mathbb{R}^\infty$ you mean $c_{00}$, the space of all real sequences with at most finitely many nonzero terms, then you can construct an inner product similarly like in $\mathbb{R}^n$:

For $x = (x_n)_{n=1}^\infty, y = (x_n)_{n=1}^\infty \in\mathbb{R}^\infty$ define

$$\langle x, y\rangle = \sum_{n=1}^\infty x_ny_n$$

This sum is finite as at most finitely many $x_ny_n$ are nonzero so $\langle\cdot, \cdot\rangle$ is well defined. You can check that it is an inner product on $\mathbb{R}^\infty$.

If $\mathbb{R}^\infty = \mathbb{R}^\mathbb{N}$, i.e. the space of all real sequences, then you cannot define it like in $c_{00}$ because the sum need not converge, but the answer is still yes.

In fact, any vector space $V$ admits an inner product.

Let $\{b_i\}_{i\in I}$ be a Hamel basis for $V$. For every $x,y \in V$ we have $x = \sum_{i \in S} \alpha_ib_i$ and $y = \sum_{j \in T} \beta_jb_j$ for some finite subsets $S, T \subseteq I$, where $\alpha_i, \beta_j \ne 0$ (the representations of this form are unique). Also set $\alpha_i = 0, \forall i \in T\setminus S$ and $\beta_j = 0, \forall j \in S\setminus T$.

Now it makes sense to define: $$\langle x, y\rangle = \sum_{k \in S \cup T} \alpha_k\overline{\beta_k}$$

You can check that $\langle\cdot, \cdot\rangle$ is an inner product on $V$.

Answer 2

The short answer is, yes. But you may need to get a little bit creative in your definitions.

For example, the set of continuous functions $f:[0,1]\to \mathbb R$ forms a vector space in $\mathbb R^\infty$

$\int_0^1 f(x)g(x) \ dx$ is an inner product for this space.

Answer 3

Yes, with some additional criteria on the inner product. If you try to extend the inner product on $\mathbb{R}^n$ for example to $\mathbb{R}^\infty$ as $\langle x,y\rangle=\sum_{i=0}^\infty x_iy_i$ you run into the problem that this series may not converge. If we consider the subset of $\mathbb{R}^{\infty}$ consisting of elements where the series $\sum_{i=0}^\infty x_i^2$ converges, then on these elements the inner product is well defined and we get an inner product space (in fact we get a Hilbert space). You can look into $\ell^2$ space for more reading.