The set of all polynomials is a vector space.

**Theorem:** The set of all polynomials with coefficients in a field \(\mathbb{F}\), denoted \(P\) or \(\mathbb{F}[x]\), is a vector space over \(\mathbb{F}\).

- **Addition:** \((a_0 + a_1x + \cdots)(b_0 + b_1x + \cdots) = (a_0+b_0) + (a_1+b_1)x + \cdots\)
- **Scalar multiplication:** \(c(a_0 + a_1x + a_2x^2 + \cdots) = (ca_0) + (ca_1)x + (ca_2)x^2 + \cdots\)

**Zero vector:** The zero polynomial \(p(x) = 0\).

This space is *infinite-dimensional*. The set \(\{1, x, x^2, x^3, \ldots\}\) is an infinite basis.

**Note:** Polynomials must have finitely many nonzero terms (each individual polynomial has finite degree), but there is no upper bound on the degree across the whole space.Copy