Sequence Subspaces From Coordinate Constraints

Coordinate Conditions on Sequences

A sequence can be viewed as a vector with one coordinate for each positive integer. This makes coordinate conditions a natural way to describe subsets of a sequence space: we might require one particular term to vanish, or require all but finitely many terms to vanish. The key question is whether the condition survives the operations of the ambient space. A zero-coordinate condition does; fixing a coordinate to a nonzero value does not.

Throughout, the scalars are real, \(\mathbb{N}=\{1,2,3,\dots\}\), and the ambient vector space is \(\mathbb{R}^{\mathbb{N}}\), the space of all real sequences \(\mathbf{x}=(x_1,x_2,x_3,\dots)\). Addition and scalar multiplication are coordinatewise: if \(\mathbf{x}=(x_n)\), \(\mathbf{y}=(y_n)\), and \(a\in\mathbb{R}\), then \(\mathbf{x}+\mathbf{y}=(x_n+y_n)\) and \(a\mathbf{x}=(ax_n)\). Its zero vector is the sequence \(\mathbf{0}=(0,0,0,\dots)\).

For sequences, closure can often be checked by looking at each coordinate. If every sequence in a set has a zero in coordinate \(k\), then adding two such sequences keeps coordinate \(k\) zero, and multiplying one by any scalar does too. This simple observation gives a general result.

Proof. The zero sequence belongs to \(Z_J\), since its \(j\)th coordinate is \(0\) for every \(j\in J\). Let \(\mathbf{x},\mathbf{y}\in Z_J\), and let \(a\in\mathbb{R}\). For each \(j\in J\), the defining condition gives \(x_j=y_j=0\). Therefore \((\mathbf{x}+\mathbf{y})_j=x_j+y_j=0+0=0\), and \((a\mathbf{x})_j=ax_j=a\cdot 0=0\). Thus both \(\mathbf{x}+\mathbf{y}\) and \(a\mathbf{x}\) belong to \(Z_J\), so the subspace test is satisfied. If \(J\) is empty, the condition imposes no restrictions and \(Z_J=\mathbb{R}^{\mathbb{N}}\), which is also consistent with the proof. \(\square\)

Taking \(J=\{k\}\) gives the set of sequences with a specified zero term. Taking several indices in \(J\) imposes several zero-coordinate conditions at once. The proof works regardless of how many indices are specified, because it verifies the condition separately at each one.

One Specified Zero Term

For a fixed \(k\in\mathbb{N}\), write \(Z_k=\{\mathbf{x}\in\mathbb{R}^{\mathbb{N}}:x_k=0\}\). This condition does not restrict the other coordinates: they may be zero or nonzero, and there may be infinitely many nonzero terms. It requires exactly that the \(k\)th term vanish.

A useful distinction is that a specified zero term does not mean a sequence has only finitely many nonzero terms. For example, the sequence \((1,1,1,0,1,1,\dots)\), with its fourth term set to zero and every other term equal to \(1\), lies in \(Z_4\) and has infinitely many nonzero terms. Finite support is a stronger condition.

Sequences with Finite Support

For instance, \((2,0,-1,0,0,\dots)\) has support \(\{1,3\}\), so it belongs to \(c_{00}\). In contrast, \((1,1,1,1,\dots)\) has support \(\mathbb{N}\) and does not belong to \(c_{00}\). The set \(c_{00}\) is a subset of \(\mathbb{R}^{\mathbb{N}}\), but it contains only sequences that are eventually zero.

Proof. The zero sequence has empty support, which is finite, so \(\mathbf{0}\in c_{00}\). Let \(\mathbf{x},\mathbf{y}\in c_{00}\). If the \(n\)th coordinate of \(\mathbf{x}+\mathbf{y}\) is nonzero, then at least one of \(x_n\) and \(y_n\) is nonzero. Consequently, \(\operatorname{supp}(\mathbf{x}+\mathbf{y})\subseteq \operatorname{supp}\mathbf{x}\cup\operatorname{supp}\mathbf{y}\). The right-hand side is a union of two finite sets and is therefore finite. Thus \(\mathbf{x}+\mathbf{y}\in c_{00}\). Now let \(a\in\mathbb{R}\). If \(a=0\), then \(a\mathbf{x}=\mathbf{0}\), which has finite support. If \(a\ne0\), then \(ax_n\ne0\) exactly when \(x_n\ne0\), so \(\operatorname{supp}(a\mathbf{x})=\operatorname{supp}\mathbf{x}\), which is finite. Hence \(a\mathbf{x}\in c_{00}\) for every scalar \(a\), and the subspace test proves the result. \(\square\)

When a Fixed Coordinate Is Nonzero

Now fix an index \(k\in\mathbb{N}\) and a real number \(c\), and consider the set \(C_{k,c}=\{\mathbf{x}\in\mathbb{R}^{\mathbb{N}}:x_k=c\}\). When \(c=0\), this is exactly \(Z_k\), so it is a subspace. When \(c\ne0\), it cannot be a subspace because it does not contain the zero sequence: the \(k\)th coordinate of \(\mathbf{0}\) is \(0\), not \(c\).

The same reasoning applies to any \(c\ne0\): if \(\mathbf{x}\in C_{k,c}\), then \(0\mathbf{x}=\mathbf{0}\), and \(\mathbf{0}\notin C_{k,c}\). The distinction is the value prescribed by the condition. Requiring a coordinate to be zero preserves the zero vector and is compatible with scalar multiplication; requiring it to equal a fixed nonzero number is not.

Common Mistake: Confusing a Zero Condition with a Fixed Value

A tempting but incorrect argument is: “The third coordinate stays fixed at \(2\), so adding two sequences with third coordinate \(2\) keeps the condition.” The calculation shows otherwise. If \(x_3=2\) and \(y_3=2\), then \((\mathbf{x}+\mathbf{y})_3=x_3+y_3=2+2=4\), not \(2\). For example, adding \((1,0,2,0,\dots)\) to itself gives \((2,0,4,0,\dots)\), which is not in \(C_{3,2}\). Scalar multiplication also fails: multiplying by \(3\) changes the third coordinate from \(2\) to \(6\).

For the homogeneous condition \(x_3=0\), the corresponding calculations are \(0+0=0\) and \(a\cdot0=0\) for every real \(a\). That is why the zero-coordinate set passes the subspace test. The value \(c=0\) is not just a convenient special case; it is what makes the condition compatible with the required operations.

Why These Conditions Are Useful

Coordinate descriptions let us identify subspaces of an infinite-dimensional ambient space without needing to list their elements one by one. A zero-coordinate requirement describes a set that is closed under every linear combination: at each constrained index, the resulting coordinate remains zero. Finite support imposes a different kind of restriction. Although the location of the nonzero terms may vary from sequence to sequence, the union of the supports of two sequences is still finite, which is the essential closure argument for addition.

These two subspaces also illustrate different patterns. A sequence can satisfy a specified zero-coordinate condition while having infinitely many nonzero terms. A sequence in \(c_{00}\) must instead have only finitely many nonzero terms, but no particular coordinate is required to vanish. When checking any coordinate-defined set, first identify exactly what is constrained, then test the zero sequence, addition, and scalar multiplication against that condition.