Linear Spaces

Definition: A Linear Space (or Vector Space) over $\mathbb{R}$ (or over $\mathbb{C}$) is a set $X$ with the binary operation of addition on elements of $X$, and a well-defined operation of scalar multiplication $\cdot$ on numbers in $\mathbb{R}$ (or $\mathbb{C}$) with elements in $X$, such that the following properties are satisfied: 1) If $x, y \in X$ then $(x + y) \in X$ (Closure under $+$). 2) $x + (y + z) = (x + y) + z$ for all $x, y, z \in X$ (Associativity of $+$). 3) $x + y = y + x$ for all $x, y \in X$ (Commutativity of $+$. 4) There exists an element $0 \in X$ such that $x + 0 = 0 = x = x$ (Existence of an additive identity). 5) For all $x \in X$ there exists an element $-x \in X$ such that $x + (-x) = (-x) + x = 0$ (Existence of additive inverses). 6) For all $\alpha \in \mathbb{R}$ (or $\mathbb{C}$), if $x \in X$ then $\alpha x \in X$ (Closure under $\cdot$). 7) $1x = x1 = x$ for all $x \in X$. 8) For all $\alpha, \beta \in \mathbb{R}$ (or $\mathbb{C}$), if $x \in X$ then $\alpha (\beta x) = (\alpha \beta)x$. 9) For all $\alpha \in \mathbb{R}$ (or $\mathbb{C}$), if $x, y \in X$ then $\alpha (x + y) = \alpha x + \alpha y$. 10) For all $\alpha, \beta \in \mathbb{R}$, if $x \in X$ then $(\alpha + \beta) x = \alpha x + \beta x$.

Let $A \subseteq \mathbb{R}$ and let $X$ be the set of functions $f : A \to \mathbb{R}$. With the binary operation $+$ of function addition defined for all $f + g \in X$ by:

(1)

\begin{align} \quad (f + g)(x) = f(x) + g(x) \end{align}

And scalar multiplication $\cdot$ defined for all $\alpha \in \mathbb{R}$ and for all $f \in X$ by:

(2)

\begin{align} \quad (\alpha f)(x) = \alpha f(x) \end{align}

Then the set $X$ with these operations forms a linear space.

Definition: If $X$ is a linear space over $\mathbb{R}$ (or $\mathbb{C}$) with the operations of addition and scalar multiplication and if $Y \subseteq X$ then $Y$ is a Linear Subspace (or Vector Subspace) of $X$ if $Y$ is a linear space with the operations of addition and scalar multiplication.

We now state a relatively simple criterion for determining whether a subset $Y$ of a linear space $X$ is a linear subspace or not.

Theorem 1: Let $X$ be a linear space over $\mathbb{R}$ (or $\mathbb{C}$) and let $Y \subseteq X$. Then $Y$ is a linear subspace of $X$ if: 1) For all $x, y \in Y$ we have that $(x + y) \in Y$. 2) For all $\alpha \in \mathbb{R}$ (or $\mathbb{C}$), if $x \in Y$ then $\alpha x \in Y$. 3) $0 \in Y$.

• Proof: We are given that (1), (4), and (6) from the definition of a linear space are given from the hypotheses of the theorem.

• Properties (2), (3), (7), (8), (9), and (10) from the definition of a linear space $X$ are all inherited properties to $Y$ from the operations of addition and scalar multiplication.

• All that remains to show is that (5) holds. Let $x \in Y$. Then since $-1 \in \mathbb{R}$ (or $\mathbb{C}$) we have that $-1x = -x \in Y$. So indeed (5) holds.

• Hence $Y$ is a linear subspace of $X$. $\blacksquare$