5  Polynomial Spaces and Infinite Dimensions

5.1 Motivation: Beyond Finite Coordinates

In Calculus I (§1.7), we encountered vectors in \mathbb{R}^2 as arrows in the plane, each uniquely determined by two coordinates (x, y). We learned that any vector can be written as a linear combination of basis vectors: \mathbf{v} = x\mathbf{e}_1 + y\mathbf{e}_2.

The dimension of the space—in this case, two—counts the number of independent directions. Every vector is specified by exactly two numbers, no more and no fewer.

But polynomials suggest a different possibility. Consider the space of all polynomials of degree at most three: p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3.

Each polynomial is determined by four coefficients (a_0, a_1, a_2, a_3). The functions \{1, x, x^2, x^3\} play the role of basis vectors, and the coefficients are coordinates. This space has dimension four.

What happens if we remove the degree restriction? The space of all polynomials—without any bound on degree—requires infinitely many basis functions \{1, x, x^2, x^3, \ldots\}. Each polynomial uses only finitely many of these, but there is no uniform bound: we need the entire infinite collection to describe all polynomials.

This chapter develops the theory of such infinite-dimensional vector spaces. Our primary examples will be spaces of polynomials and power series, which form the foundation for representing functions as infinite sums. The framework developed here will allow us to understand power series as infinite linear combinations in a basis. Just as polynomials are finite sums a_0 + a_1 x + \cdots + a_n x^n, power series extend this to infinite sums \sum_{n=0}^{\infty} c_n x^n. The question of when such infinite combinations converge—answered by the theory of series from previous sections—determines where these expressions define functions.

5.2 Vector Spaces: The Axioms

We begin by stating precisely what constitutes a vector space. The definition abstracts the essential properties of \mathbb{R}^2: we can add vectors and scale them by real numbers, and these operations satisfy natural algebraic laws.

Definition 5.1 (Vector Space) A vector space over \mathbb{R} is a set V together with two operations—addition and scalar multiplication—satisfying:

  1. Closure under addition: \mathbf{u} + \mathbf{v} \in V
  2. Commutativity: \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}
  3. Associativity: (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})
  4. Zero vector exists: there is \mathbf{0} with \mathbf{v} + \mathbf{0} = \mathbf{v}
  5. Additive inverse: for each \mathbf{v} there exists -\mathbf{v} with \mathbf{v} + (-\mathbf{v}) = \mathbf{0}
  6. Closure under scalar multiplication: a \mathbf{v} \in V
  7. Distributivity over vector addition: a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}
  8. Distributivity over scalar addition: (a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}
  9. Associativity of scalars: a(b\mathbf{v}) = (ab)\mathbf{v}
  10. Multiplicative identity: 1 \mathbf{v} = \mathbf{v}

Elements of V are called vectors, and elements of \mathbb{R} are called scalars.

These axioms make no reference to geometry, arrows, or coordinates. They specify only how objects combine algebraically. Any collection of objects satisfying these axioms—whether vectors in \mathbb{R}^2, functions, sequences, or polynomials—inherits the same algebraic structure.

Example 5.1 (\mathbb{R}^2 as a Vector Space) The set \mathbb{R}^2 = \{(x, y) : x, y \in \mathbb{R}\} with componentwise addition and scalar multiplication forms a vector space. We add vectors by adding coordinates: (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2). We scale by multiplying each coordinate: c(x, y) = (cx, cy). The zero vector is (0, 0), and the additive inverse of (x, y) is (-x, -y). Each axiom can be verified directly from properties of real numbers.

Example 5.2 (Functions as a Vector Space) The set of all functions f : \mathbb{R} \to \mathbb{R} forms a vector space under pointwise operations. We add functions by (f + g)(x) = f(x) + g(x) and scale by (cf)(x) = c \cdot f(x). The zero vector is the constant function 0(x) = 0 for all x. Each axiom follows from properties of function evaluation and real number arithmetic.

From these examples, we extract a pattern: vector spaces consist of objects that can be added and scaled, with the operations behaving like ordinary arithmetic. The abstraction allows us to study polynomials, functions, sequences, and other mathematical objects using the same conceptual framework developed for geometric vectors.

5.3 Polynomial Spaces

Our primary concern is spaces of polynomials. These spaces illustrate the transition from finite to infinite dimensions while remaining algebraically tractable.

Definition 5.2 (Space of Polynomials of Bounded Degree) For n \in \mathbb{N}, the set \mathcal{P}_n = \left\{p(x) = a_0 + a_1 x + \cdots + a_n x^n : a_i \in \mathbb{R}\right\} denotes the space of all polynomials of degree at most n.

Addition and scalar multiplication are defined in the natural way: if p(x) = \sum_{i=0}^{n} a_i x^i and q(x) = \sum_{i=0}^{n} b_i x^i, then (p + q)(x) = \sum_{i=0}^{n} (a_i + b_i) x^i and (cp)(x) = \sum_{i=0}^{n} (ca_i) x^i. These operations combine polynomials by combining their coefficients.

Verification of the vector space axioms reduces to verifying properties of coefficient arithmetic. For instance, commutativity of addition follows from a_i + b_i = b_i + a_i for each coefficient. The zero vector is the zero polynomial 0(x) = 0 (all coefficients zero), and the additive inverse of p(x) = \sum a_i x^i is -p(x) = \sum (-a_i) x^i.

Example 5.3 (Operations in \mathcal{P}_2) The space \mathcal{P}_2 consists of all quadratic polynomials a_0 + a_1 x + a_2 x^2. The polynomial p(x) = 3 + 2x - x^2 lies in \mathcal{P}_2, as does q(x) = 1 - 4x + 5x^2. Their sum is p(x) + q(x) = 4 - 2x + 4x^2, which also belongs to \mathcal{P}_2. Scaling p by c = -2 gives -6 - 4x + 2x^2 \in \mathcal{P}_2.

Definition 5.3 (Space of All Polynomials) The set \mathcal{P} = \mathcal{P}_0\cup \mathcal{P}_1 \cup \mathcal{P}_2 \cup \ldots = \bigcup_{n=0}^{\infty}\mathcal{P}_n denotes the space of all polynomials (of any degree).

Every element of \mathcal{P} is a polynomial of some finite degree, but the space itself contains polynomials of arbitrarily high degree. The distinction between \mathcal{P}_n and \mathcal{P} is fundamental. Each polynomial in \mathcal{P} has some finite degree, but there is no uniform bound: the space contains polynomials of degree 1, 2, 3, \ldots without limit. This makes \mathcal{P} infinite-dimensional.

The vector space operations extend naturally to \mathcal{P}. If p has degree m and q has degree n with m \leq n, we may add them by treating p as having degree n with zero coefficients for terms of degree greater than m. The sum has degree at most \max(m, n), hence lies in \mathcal{P}.

5.4 Linear Combinations and Span

In \mathbb{R}^2, every vector can be written as x\mathbf{e}_1 + y\mathbf{e}_2—a linear combination of basis vectors. This property extends to arbitrary vector spaces.

Definition 5.4 (Linear Combination) Let V be a vector space and \mathbf{v}_1, \ldots, \mathbf{v}_k \in V. A linear combination of these vectors is any expression of the form c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_k \mathbf{v}_k where c_1, \ldots, c_k \in \mathbb{R} are scalars.

Linear combinations generate new vectors from old ones. The set of all possible linear combinations forms a subspace—a subset closed under the vector space operations.

Definition 5.5 (Span) Let S = \{\mathbf{v}_1, \ldots, \mathbf{v}_k\} be a finite set of vectors in V. The span of S is the set of all linear combinations: \text{span}(S) = \left\{\sum_{i=1}^k c_i \mathbf{v}_i : c_i \in \mathbb{R}\right\}.

If \text{span}(S) = V, we say S spans V, or that S is a spanning set for V.

The span captures “everything reachable” by linear combinations. In \mathbb{R}^2, the span of \{\mathbf{e}_1, \mathbf{e}_2\} is all of \mathbb{R}^2, since every vector (x, y) = x\mathbf{e}_1 + y\mathbf{e}_2 is a linear combination. The span of a single nonzero vector \mathbf{v} is the line through the origin in the direction of \mathbf{v}.

Example 5.4 (Spanning \mathcal{P}_2) In \mathcal{P}_2, consider S = \{1, x, x^2\}. Every polynomial a_0 + a_1 x + a_2 x^2 is a linear combination a_0 \cdot 1 + a_1 \cdot x + a_2 \cdot x^2, so \text{span}(S) = \mathcal{P}_2. The set S spans \mathcal{P}_2.

Example 5.5 (Incomplete Spanning Set) In \mathcal{P}_2, let T = \{1, x\}. The span consists of all polynomials a_0 + a_1 x—polynomials of degree at most one. We have \text{span}(T) = \mathcal{P}_1 \subsetneq \mathcal{P}_2. The set T does not span \mathcal{P}_2 since x^2 \notin \text{span}(T).

Example 5.6 (Spanning \mathcal{P}) In \mathcal{P}, the set \{1, x, x^2, x^3, \ldots\} spans the space. Every polynomial has finite degree n, hence is a linear combination of \{1, x, \ldots, x^n\}. The span includes all finite linear combinations—this is the entire space \mathcal{P}.

The span always forms a subspace, as the next result confirms.

Theorem 5.1 (Span is a Subspace) Let S = \{\mathbf{v}_1, \ldots, \mathbf{v}_k\} be a set of vectors in V. Then \text{span}(S) is a subspace of V.

We verify closure under addition and scalar multiplication.

Let \mathbf{u}, \mathbf{w} \in \text{span}(S). Then there exist scalars a_i, b_i such that \mathbf{u} = \sum_{i=1}^{k} a_i \mathbf{v}_i, \quad \mathbf{w} = \sum_{i=1}^{k} b_i \mathbf{v}_i.

Their sum is \mathbf{u} + \mathbf{w} = \sum_{i=1}^{k} (a_i + b_i) \mathbf{v}_i \in \text{span}(S).

For any scalar c, c\mathbf{u} = c\sum_{i=1}^{k} a_i \mathbf{v}_i = \sum_{i=1}^{k} (ca_i) \mathbf{v}_i \in \text{span}(S).

The zero vector is \mathbf{0} = 0\mathbf{v}_1 + \cdots + 0\mathbf{v}_k \in \text{span}(S). Thus \text{span}(S) is closed under the required operations and contains the zero vector, confirming it is a subspace. \square

5.5 Linear Independence

Not all spanning sets are equally efficient. In \mathbb{R}^2, the set \{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_1 + \mathbf{e}_2\} spans the plane, but the third vector is redundant: it adds no new directions. We formalize this notion through linear independence.

Definition 5.6 (Linear Independence) A finite set of vectors \{\mathbf{v}_1, \ldots, \mathbf{v}_k\} in V is linearly independent if the only linear combination producing the zero vector is the trivial one: c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k = \mathbf{0} \implies c_1 = \cdots = c_k = 0.

If the set is not linearly independent, it is linearly dependent.

Linear dependence means one vector can be written as a combination of the others, introducing redundancy. Linear independence ensures no such redundancy exists—each vector contributes an essential new direction.

Example 5.7 (Linear Independence in \mathcal{P}_2) In \mathcal{P}_2, the set \{1, x, x^2\} is linearly independent. Suppose a_0 \cdot 1 + a_1 \cdot x + a_2 \cdot x^2 = 0 (the zero polynomial). Then a_0 + a_1 x + a_2 x^2 = 0 for all x. Evaluating at distinct points: at x = 0, we get a_0 = 0; differentiating yields a_1 + 2a_2 x = 0 for all x, forcing a_1 = a_2 = 0. Thus a_0 = a_1 = a_2 = 0, confirming independence.

Example 5.8 (Linear Dependence in \mathcal{P}_2) In \mathcal{P}_2, the set \{1, x, 2 + 3x\} is linearly dependent. Observe that 2 \cdot 1 + 3 \cdot x - 1 \cdot (2 + 3x) = 0 with nonzero coefficients (2, 3, -1).

Theorem 5.2 (Unique Representation) Let \{\mathbf{v}_1, \ldots, \mathbf{v}_k\} be linearly independent. If \mathbf{w} \in \text{span}(\mathbf{v}_1, \ldots, \mathbf{v}_k), then the coefficients in the representation \mathbf{w} = c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k are uniquely determined.

Suppose \mathbf{w} = c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k and also \mathbf{w} = d_1 \mathbf{v}_1 + \cdots + d_k \mathbf{v}_k. Subtracting gives \mathbf{0} = (c_1 - d_1)\mathbf{v}_1 + \cdots + (c_k - d_k)\mathbf{v}_k.

By linear independence, c_i - d_i = 0 for each i, hence c_i = d_i. \square

This uniqueness justifies calling the coefficients “coordinates”: they provide an unambiguous numerical description of the vector relative to the spanning set.

5.6 Bases and Dimension

A basis combines the efficiency of linear independence with the completeness of spanning.

Definition 5.7 (Basis) A set \mathcal{B} = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} is a basis for V if:

  1. \mathcal{B} spans V, and

  2. \mathcal{B} is linearly independent.

Equivalently, \mathcal{B} is a basis if every vector in V can be written uniquely as a linear combination of vectors in \mathcal{B}.

In finite-dimensional spaces, all bases have the same size—a fact we now prove.

Theorem 5.3 (Invariance of Basis Size) If V has a finite basis, then every basis of V has the same number of elements.

Let \mathcal{B} = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} and \mathcal{C} = \{\mathbf{w}_1, \ldots, \mathbf{w}_m\} be two bases for V. We show m = n by proving m \leq n and n \leq m.

Since \mathcal{B} spans V, each \mathbf{w}_j can be written as \mathbf{w}_j = \sum_{i=1}^{n} a_{ij} \mathbf{v}_i.

Suppose m > n. Consider the linear combination \sum_{j=1}^{m} c_j \mathbf{w}_j = \sum_{j=1}^{m} c_j \left(\sum_{i=1}^{n} a_{ij} \mathbf{v}_i\right) = \sum_{i=1}^{n} \left(\sum_{j=1}^{m} a_{ij} c_j\right) \mathbf{v}_i.

Setting this equal to \mathbf{0} and using independence of \mathcal{B} gives the system \sum_{j=1}^{m} a_{ij} c_j = 0 \quad \text{for } i = 1, \ldots, n.

This is a homogeneous system of n equations in m unknowns. Since m > n, the system has a nontrivial solution (c_1, \ldots, c_m) with some c_j \neq 0. But then \sum_{j=1}^{m} c_j \mathbf{w}_j = \mathbf{0} with nonzero coefficients, contradicting independence of \mathcal{C}.

Thus m \leq n. By symmetry, n \leq m, hence m = n. \square

This invariance justifies the following definition.

Definition 5.8 (Dimension) If V has a finite basis of n elements, the dimension of V is n, written \dim(V) = n.

Example 5.9 (Dimension of \mathcal{P}_n) The space \mathcal{P}_n has dimension n + 1. The set \{1, x, x^2, \ldots, x^n\} is a basis: it spans \mathcal{P}_n by definition of the space, and linear independence follows from the fact that a nonzero polynomial of degree at most n has at most n roots, hence cannot vanish identically. Any other basis for \mathcal{P}_n must also contain exactly n + 1 elements.

5.7 The Infinite-Dimensional Case

The space \mathcal{P} of all polynomials does not have a finite basis. To see this, suppose \mathcal{B} = \{p_1, \ldots, p_k\} were a finite spanning set. Each p_i has some degree d_i, and the maximum degree among them is d = \max\{d_1, \ldots, d_k\}. Any linear combination of the p_i has degree at most d. But the polynomial x^{d+1} has degree d + 1 and thus cannot be written as a linear combination of \mathcal{B}. This contradicts the assumption that \mathcal{B} spans \mathcal{P}.

The space \mathcal{P} does have a basis, but it is infinite. To formalize infinite-dimensional spaces, we must specify what “basis” means when infinitely many vectors are involved. The key restriction is that linear combinations remain finite—we do not sum infinitely many terms. That requires convergence of series, which we address separately.

Definition 5.9 (Infinite Basis) An infinite set \mathcal{B} is a basis for V if:

  1. Every element of V can be written as a finite linear combination of elements from \mathcal{B}, and

  2. No element of \mathcal{B} lies in the span of the others.

A vector space with an infinite basis is infinite-dimensional.

The requirement that linear combinations remain finite is essential. We do not yet have the machinery to discuss infinite sums—that requires the theory of series developed elsewhere in this text.

Example 5.10 (Basis for \mathcal{P}) The set \mathcal{B} = \{1, x, x^2, x^3, \ldots\} is a basis for \mathcal{P}. Every polynomial has finite degree, hence is a finite linear combination of elements from \mathcal{B}. The set is linearly independent: if a_0 + a_1 x + \cdots + a_n x^n = 0 for some finite n, then all coefficients must vanish, as this polynomial is identically zero.

Theorem 5.4 (Dimensional Hierarchy) If U is a subspace of V with \dim(U) = \dim(V) and both dimensions are finite, then U = V.

Let \dim(U) = \dim(V) = n, and let \mathcal{B} = \{\mathbf{u}_1, \ldots, \mathbf{u}_n\} be a basis for U. Since \mathcal{B} \subset U \subset V and \mathcal{B} is linearly independent in U, it remains linearly independent in V.

If \mathcal{B} spans V, then U = V by definition of U as the span of its basis. Suppose, for contradiction, that \mathcal{B} does not span V. Then there exists \mathbf{v} \in V \setminus \text{span}(\mathcal{B}). The set \{\mathbf{u}_1, \ldots, \mathbf{u}_n, \mathbf{v}\} is linearly independent: any dependence relation would express \mathbf{v} as a combination of the \mathbf{u}_i, contradicting \mathbf{v} \notin \text{span}(\mathcal{B}).

This gives a linearly independent set of n + 1 vectors in V, contradicting \dim(V) = n. Thus \mathcal{B} spans V, hence U = V. \square

This result has no analog in infinite dimensions. The space \mathcal{P} has proper subspaces (such as the span of \{x, x^2, x^3, \ldots\}, excluding constant polynomials) that are also infinite-dimensional. The notion of “size” becomes more subtle when infinitely many basis vectors are involved.

5.8 Polynomial Spaces and Power Series

The space \mathcal{P} consists of finite linear combinations in the basis \{1, x, x^2, \ldots\}. Each polynomial uses finitely many basis elements—say, 1, x, \ldots, x^n—with all higher coefficients zero.

Power series remove this restriction. A power series f(x) = \sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + \cdots allows infinitely many nonzero coefficients. The coordinate vector (c_0, c_1, c_2, \ldots) is an infinite sequence. Whether this defines a function depends on whether the series converges at a given x—a question answered by the theory developed in previous sections.

Such expressions do not automatically define elements of a vector space. The series must converge, and convergence depends on x. The machinery of infinite series, developed in the preceding sections of this chapter, provides the necessary framework for determining when such infinite sums exist.

A power series with radius of convergence R > 0 defines a function on the interval (-R, R). For each fixed x in this interval, the infinite sum \sum c_n x^n converges to a real number, producing a function value f(x). The set of all such functions—those representable as convergent power series—forms a vector space under pointwise addition and scalar multiplication.

This space is still infinite-dimensional, but it is larger than \mathcal{P}. Functions like e^x, \sin(x), and \frac{1}{1-x} have power series representations but are not polynomials. The space of analytic functions (those equal to their power series on some interval) strictly contains the space of polynomials.

The vector space structure connects directly to the algebraic properties of series:

  • The sum of two convergent power series is a power series (with radius the minimum of the two radii)

  • Scalar multiples of convergent power series are power series

  • The zero function corresponds to the series with all coefficients zero

5.9 Coordinates and the Standard Basis

Once a basis is fixed, every vector acquires coordinates. In \mathcal{P}_n with the standard basis \{1, x, x^2, \ldots, x^n\}, the polynomial p(x) = a_0 + a_1 x + \cdots + a_n x^n has coordinate vector (a_0, a_1, \ldots, a_n) \in \mathbb{R}^{n+1}. This establishes a one-to-one correspondence between \mathcal{P}_n and \mathbb{R}^{n+1}. This correspondence preserves addition and scalar multiplication: adding polynomials corresponds to adding coordinate vectors componentwise; scaling a polynomial corresponds to scaling its coordinate vector. The algebraic structure of \mathcal{P}_n is identical to that of \mathbb{R}^{n+1}.

In infinite-dimensional spaces, the concept of coordinates extends formally, but practical considerations intervene. A power series \sum c_n x^n can be viewed as having coordinates (c_0, c_1, c_2, \ldots)—an infinite sequence. The space of such sequences, under componentwise addition and scalar multiplication, is itself a vector space (often denoted \mathbb{R}^{\mathbb{N}}). Convergence of the power series imposes additional structure: not all sequences of coefficients produce convergent series. The radius of convergence formula (ratio or root test) determines which sequences correspond to well-defined functions.

5.10 Summary and Forward Connections

We have established that polynomials and power series inhabit vector spaces—abstract structures defined by addition and scalar multiplication satisfying algebraic axioms. The space \mathcal{P}_n has dimension n + 1 and admits bases like \{1, x, \ldots, x^n\}. The space \mathcal{P} of all polynomials is infinite-dimensional, with basis \{1, x, x^2, \ldots\} where every polynomial is a finite linear combination of basis elements.

Power series extend this framework by allowing infinite linear combinations. Convergence of the series depends on the coefficients and the value of x. The space of functions represented by convergent power series is infinite-dimensional and strictly larger than \mathcal{P}.

These vector spaces provide the setting for Taylor series. A smooth function f \in C^\infty (see smoothness classes) has derivatives f^{(k)}(a) at a point a. These derivatives serve as coordinates in the normalized basis \left\{1, (x-a), \frac{(x-a)^2}{2!}, \frac{(x-a)^3}{3!}, \ldots\right\}. The Taylor series \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k is the infinite linear combination with these coordinates. Whether this series converges to f(x) depends on the function’s behavior—specifically, on how rapidly the remainder R_n(x) = f(x) - T_n(x) approaches zero. Functions for which the Taylor series converges to the function on some interval are called analytic, and they form a subspace of the space of smooth functions C^{\infty}.

Problems

  1. Verify that \mathcal{P}_3 satisfies the vector space axioms.

  2. Show that \{1, x, x^2\} is linearly independent in \mathcal{P}_2.

  3. Find the coordinates of p(x) = 2 + 3x - x^2 in the basis \{1, (x-1), (x-1)^2\} for \mathcal{P}_2.

  4. Prove that \{1, x, x+1\} is linearly dependent in \mathcal{P}_2.

  5. Show that \text{span}\{1, x\} = \mathcal{P}_1.