The subspace $\Gamma$ of directions along which a series converges absolutely, and its orthogonal complement $\Gamma^{\perp}$

Let $n \in \mathbb{N}$ with $n \ge 1$ and let $(x_k)$ be a sequence in $\mathbb{R}^{n}$ (Series of vectors in $\mathbb{R}^n$, absolute convergence, rearrangement, and the set of rearrangement sums). Define

$\Gamma \;:=\; \Bigl\{\, a \in \mathbb{R}^{n} \;:\; \sum_k \bigl|\langle a, x_k\rangle\bigr| \text{ converges} \,\Bigr\}, \qquad \Gamma^{\perp} \;:=\; \bigl\{\, y \in \mathbb{R}^{n} \;:\; \langle a, y\rangle = 0 \text{ for every } a \in \Gamma \,\bigr\},$

the inner product being the Euclidean one (The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$) and the series that of Series, partial sums, convergence and the sum, divergence, and the tail series. Elements of $\Gamma$ are the summing directions of $(x_k)$: those $a$ for which the real series of the projections $\langle a, x_k\rangle$ converges absolutely (Absolutely convergent and conditionally convergent series, and the general starting index). Both sets depend on the sequence $(x_k)$; when several are in play the notation is $\Gamma(x)$ and $\Gamma(x)^{\perp}$.

Phrased with the inner product, deliberately. Abstract linear maps are already defined in Linear map between vector spaces over the same field, so a linear functional can be read as a linear map into $\mathbb{R}$. This library does not yet define the dual space or prove that every such functional on $\mathbb{R}^{n}$ is represented by an inner product with a vector. Writing $\Gamma$ with Euclidean directions avoids presupposing that agreement, and nothing on this page depends on it.

Both are linear subspaces

$\Gamma$ is a linear subspace of $\mathbb{R}^{n}$ (Linear subspace of a vector space). It is nonempty: $\langle 0, x_k\rangle = 0$ for every $k$ by bilinearity, and the series with all terms $0$ converges. For $\lambda \in \mathbb{R}$ and $a, b \in \Gamma$, bilinearity and the absolute value laws give

$\bigl|\langle \lambda a + b, x_k\rangle\bigr| \;=\; \bigl|\lambda\langle a,x_k\rangle + \langle b,x_k\rangle\bigr| \;\le\; |\lambda|\,\bigl|\langle a,x_k\rangle\bigr| + \bigl|\langle b,x_k\rangle\bigr|$

(Basic properties of the absolute value), and the series of the right-hand side converges by Convergent series add and scale termwise clauses 1 and 2, so the left-hand series converges by the comparison test (If $0 \le a_k \le b_k$ eventually, convergence of $\sum b_k$ gives convergence of $\sum a_k$, and divergence of $\sum a_k$ gives divergence of $\sum b_k$, the terms being nonnegative). By the one-step subspace test (One-step subspace test: a nonempty $W \subseteq V$ is a linear subspace if and only if $\lambda u + v \in W$ for all $\lambda \in F$ and $u, v \in W$), $\Gamma$ is a linear subspace.

$\Gamma^{\perp}$ is a linear subspace of $\mathbb{R}^{n}$. It contains $0$, and for $\lambda \in \mathbb{R}$, $y, z \in \Gamma^{\perp}$ and $a \in \Gamma$, bilinearity gives $\langle a, \lambda y + z\rangle = \lambda\langle a,y\rangle + \langle a,z\rangle = 0$; again One-step subspace test: a nonempty $W \subseteq V$ is a linear subspace if and only if $\lambda u + v \in W$ for all $\lambda \in F$ and $u, v \in W$ applies. Equivalently $\Gamma^{\perp}$ is the intersection of the linear subspaces $\{y : \langle a,y\rangle = 0\}$ over $a \in \Gamma$, a nonempty family since $0 \in \Gamma$, and The intersection of a nonempty family of linear subspaces of $V$ is a linear subspace of $V$ gives the same conclusion.

$\Gamma$ is everything exactly when the series converges absolutely

If $\sum x_k$ converges absolutely then $\Gamma = \mathbb{R}^{n}$. For any $a$, Cauchy-Schwarz gives $|\langle a,x_k\rangle| \le \lVert a\rVert_2\lVert x_k\rVert_2$ (Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$, the parallelogram law and polarisation), and $\sum_k \lVert a\rVert_2\lVert x_k\rVert_2$ converges by Convergent series add and scale termwise clause 2; the comparison test gives $a \in \Gamma$.

Conversely, if $\Gamma = \mathbb{R}^{n}$ then $\sum x_k$ converges absolutely. Each standard basis vector $e_j$ lies in $\Gamma$, and $\langle e_j, x_k\rangle = (x_k)_j$ (The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$, The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$), so each real series $\sum_k |(x_k)_j|$ converges. A finite sum of convergent series converges, by Convergent series add and scale termwise clause 1 and induction on the number of summands (The principle of mathematical induction, Laws of finite sums and finite products, Finite sums and finite products, by recursion), so $\sum_k \sum_{j<n}|(x_k)_j| = \sum_k \lVert x_k\rVert_1$ converges; and $\lVert x_k\rVert_2 \le \lVert x_k\rVert_1$ (The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$ clause 3, The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$), so $\sum_k\lVert x_k\rVert_2$ converges by the comparison test.

That equivalence is what makes the containment theorem below contain An absolutely convergent series in $\mathbb{R}^n$ converges, and every rearrangement converges to the same sum as a special case: absolute convergence gives $\Gamma = \mathbb{R}^{n}$, hence $\Gamma^{\perp} = \{0\}$ (any $y \in \Gamma^{\perp}$ satisfies $\langle y,y\rangle = 0$ and so $y = 0$ by positive definiteness), and the affine subspace below collapses to a point.

Affine subspaces

At this point in the reading order the general definition is not yet available, so the Euclidean instance is fixed here; the later Affine subspaces as translates $x+U$ of linear subspaces supplies the general definition. For a linear subspace $W \subseteq \mathbb{R}^{n}$ and $s \in \mathbb{R}^{n}$, the affine subspace through $s$ with direction $W$ is the coset

$s + W \;:=\; \{\, s + w \;:\; w \in W \,\} .$

A coset is determined by $W$ together with any one of its points. If $p \in s + W$, say $p = s + w_0$ with $w_0 \in W$, then $p + W = s + W$: every $p + w = s + (w_0 + w)$ lies in $s+W$ because $W$ is closed under addition, and every $s + w = p + (w - w_0)$ lies in $p + W$ because $W$ is closed under addition and under multiplication by $-1$ (Linear subspace of a vector space, Vector space over a field). In particular $s + W = s' + W$ if and only if $s - s' \in W$.

• $0 \in \Gamma$ always, so $\Gamma$ is never empty and $\Gamma^{\perp}$ is never larger than $\mathbb{R}^{n}$ by accident. At the other extreme, if $\Gamma = \{0\}$ then $\Gamma^{\perp} = \mathbb{R}^{n}$, the condition on $y$ being vacuous apart from $a = 0$.

• The definition does not presuppose convergence of $\sum x_k$, and neither $\Gamma$ nor $\Gamma^{\perp}$ mentions the sum. Convergence is a hypothesis of the theorems that use them, not of the definition.

• No orthogonal decomposition is claimed. Nothing here asserts that $\mathbb{R}^{n}$ is the direct sum of $\Gamma$ and $\Gamma^{\perp}$, or that $(\Gamma^{\perp})^{\perp} = \Gamma$. Those are statements of the theory of inner product spaces and orthogonality, which is planned for a page earlier in the plan order that is not yet built, and no item on this page uses them. What is used is only that $\Gamma^{\perp}$ is a linear subspace and that $\langle a, y\rangle = 0$ for $a \in \Gamma$, $y \in \Gamma^{\perp}$.

• The name. $\Gamma$ is the set of directions in which the series is absolutely summable; along a direction outside $\Gamma$ the projected real series converges conditionally at best, and it is exactly there that rearrangement can move the sum.