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

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

$\mathrm{\Gamma }:=\{a\in {\mathbb{R}}^n:{\sum }_k\mid ⟨a,x_k⟩\mid \text{ converges}\},{\mathrm{\Gamma }}^{\perp }:=\{y\in {\mathbb{R}}^n:⟨a,y⟩=0\text{ for every }a\in \mathrm{\Gamma }\},$

the inner product being the Euclidean one (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$) and the series that of Series, partial sums, convergence and the sum, divergence, and the tail series. Elements of $\mathrm{\Gamma }$ are the summing directions of $(x_k)$: those $a$ for which the real series of the projections $⟨a,x_k⟩$ 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 $\mathrm{\Gamma }(x)$ and $\mathrm{\Gamma }(x{)}^{\perp }$.

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 $\mathrm{\Gamma }$ with Euclidean directions avoids presupposing that agreement, and nothing on this page depends on it.

Both are linear subspaces

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

$\mid ⟨\lambda a+b,x_k⟩\mid =\mid \lambda ⟨a,x_k⟩+⟨b,x_k⟩\mid \le \mathrm{\mid }\lambda \mathrm{\mid }\mid ⟨a,x_k⟩\mid +\mid ⟨b,x_k⟩\mid$

(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 $u,v\in W$), $\mathrm{\Gamma }$ is a linear subspace.

${\mathrm{\Gamma }}^{\perp }$ is a linear subspace of ${\mathbb{R}}^n$. It contains $0$, and for $\lambda \in \mathbb{R}$, $y,z\in {\mathrm{\Gamma }}^{\perp }$ and $a\in \mathrm{\Gamma }$, bilinearity gives $⟨a,\lambda y+z⟩=\lambda ⟨a,y⟩+⟨a,z⟩=0$; 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 ${\mathrm{\Gamma }}^{\perp }$ is the intersection of the linear subspaces $\{y:⟨a,y⟩=0\}$ over $a\in \mathrm{\Gamma }$, a nonempty family since $0\in \mathrm{\Gamma }$, and The intersection of a nonempty family of linear subspaces of $V$ is a linear subspace of $V$ gives the same conclusion.

$\mathrm{\Gamma }$ is everything exactly when the series converges absolutely

If $\sum x_k$ converges absolutely then $\mathrm{\Gamma }={\mathbb{R}}^n$. For any $a$, Cauchy-Schwarz gives $\mathrm{\mid }⟨a,x_k⟩\mathrm{\mid }\le \parallel a{\parallel }_2\parallel x_k{\parallel }_2$ (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation), and ${\sum }_k\parallel a{\parallel }_2\parallel x_k{\parallel }_2$ converges by Convergent series add and scale termwise clause 2; the comparison test gives $a\in \mathrm{\Gamma }$.

Conversely, if $\mathrm{\Gamma }={\mathbb{R}}^n$ then $\sum x_k$ converges absolutely. Each standard basis vector $e_j$ lies in $\mathrm{\Gamma }$, and $⟨e_j,x_k⟩=(x_k{)}_j$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$), so each real series ${\sum }_k\mathrm{\mid }(x_k{)}_j\mathrm{\mid }$ 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 ${\sum }_k{\sum }_{j<n}\mathrm{\mid }(x_k{)}_j\mathrm{\mid }={\sum }_k\parallel x_k{\parallel }_1$ converges; and $\parallel x_k{\parallel }_2\le \parallel x_k{\parallel }_1$ (The finite and reverse triangle inequalities for a norm; and for $n\ge 1$ every norm $N$ on ${\mathbb{R}}^n$ satisfies $N(x)\le C\parallel x{\parallel }_1$ and is Lipschitz, hence continuous, for $d_2$ clause 3, The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$), so ${\sum }_k\parallel x_k{\parallel }_2$ 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 $\mathrm{\Gamma }={\mathbb{R}}^n$, hence ${\mathrm{\Gamma }}^{\perp }=\{0\}$ (any $y\in {\mathrm{\Gamma }}^{\perp }$ satisfies $⟨y,y⟩=0$ 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

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

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 $s+W=s^{\mathrm{\prime }}+W$ if and only if $s-s^{\mathrm{\prime }}\in W$.

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

• The definition does not presuppose convergence of $\sum x_k$, and neither $\mathrm{\Gamma }$ nor ${\mathrm{\Gamma }}^{\perp }$ 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 $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\perp }$, or that $({\mathrm{\Gamma }}^{\perp }{)}^{\perp }=\mathrm{\Gamma }$. 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 ${\mathrm{\Gamma }}^{\perp }$ is a linear subspace and that $⟨a,y⟩=0$ for $a\in \mathrm{\Gamma }$, $y\in {\mathrm{\Gamma }}^{\perp }$.

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