Linear Algebra Done Right · Problem 1C.24

Finite-Dimensional Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · direct-sums

Problem

A function f:R→Rf : \mathbf{R} \to \mathbf{R} is called even if f(−x)=f(x)f(-x) = f(x) for all x∈Rx \in \mathbf{R}. A function f:R→Rf : \mathbf{R} \to \mathbf{R} is called odd if f(−x)=−f(x)f(-x) = -f(x) for all x∈Rx \in \mathbf{R}. Let VeV_{\mathrm{e}} denote the set of real-valued even functions on R\mathbf{R} and let VoV_{\mathrm{o}} denote the set of real-valued odd functions on R\mathbf{R}. Show that RR=Ve⊕Vo.\mathbf{R}^{\mathbf{R}} = V_{\mathrm{e}} \oplus V_{\mathrm{o}}.
Intuitive explanation

Even and odd functions partition all real functions into two complementary subspaces—the key is showing that every function can be uniquely written as a sum of an even part and an odd part. Think of decomposing an arbitrary function ff by symmetry: extract what's symmetric about the yy-axis (even) and what's antisymmetric (odd), then verify these two pieces are independent and cover everything. This is a direct sum decomposition, so you need to prove both subspaces are closed under addition and scalar multiplication, and that their only overlap is the zero function.

Intuitive explanation

Even and odd functions partition all real functions into two complementary subspaces—the key is showing that every function can be uniquely written as a sum of an even part and an odd part. Think of decomposing an arbitrary function ff by symmetry: extract what's symmetric about the yy-axis (even) and what's antisymmetric (odd), then verify these two pieces are independent and cover everything. This is a direct sum decomposition, so you need to prove both subspaces are closed under addition and scalar multiplication, and that their only overlap is the zero function.

Related theory

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