The Linear Space L(X, Y)

The Linear Space L(X, Y)

Recall from the Linear Operators on Linear Spaces page that if $X$ and $Y$ are linear spaces, then a linear operator from $X$ to $Y$ is a function $T : X \to Y$ with the following properties:

  • $T(x + y) = T(x) + T(y)$ for all $x, y \in X$.
  • $T(\lambda x) = \lambda T(x)$ for all $x \in X$ and for all $\lambda \in \mathbb{C}$.

The set of linear operators from $X$ to $Y$ is denoted $\mathcal L(X, Y)$.

Definition: Let $X$ and $Y$ be linear spaces. We define the operation of Addition of Linear Operators for all $S, T \in \mathcal L(X, Y)$ by $(S + T)(x) = S(x) + T(x)$ for all $x \in X$. We define the operator of Scalar Multiplication of Linear Operators for all $T \in \mathcal L(X, Y)$ and for all $\lambda \in \mathbb{C}$ by $(\lambda T)(x) = \lambda T(x)$.

We will now show that $\mathcal L(X, Y)$ with the operations defined above is itself a linear space.

Theorem 1: Let $X$ and $Y$ be linear operators. Then $\mathcal L(X, Y)$ with the operations of function addition and scalar multiplication is a linear space.
  • Proof: Let $S, T, U \in \mathcal L(X, Y)$ and let $\alpha, \beta \in \mathbb{C}$. Then:
  • Associativity of Addition:
(1)
\begin{align} \quad \quad [S + (T + U)](x) = S(x) + (T + U)(x) = S(x) + [T(x) + U(x)] = [S(x) + T(x)] + U(x) = (S + T)(x) + U(x) = [(S + T) + U](x) \end{align}
  • Commutativity of Addition:
(2)
\begin{align} \quad (S + T)(x) = S(x) + T(x) = T(x) + S(x) = (T + S)(x) \end{align}
  • Existence of a Zero Linear Operator: Let $0 : X \to Y$ be defined for all $x \in X$ by $0(x) = 0$. Then:
(3)
\begin{align} \quad (S + 0)(x) = S(x) + 0(x) = S(x) + 0 = S(x) \end{align}
(4)
\begin{align} \quad (0 + S)(x) = 0(x) + S(x) = 0 + S(x) = S(x) \end{align}
  • Existence of Additive Inverses: For $S \in \mathcal L(X, Y)$, define $-S : X \to Y$ for all $x \in X$ by $(-S)(x) = -S(x)$. Then:
(5)
\begin{align} \quad (S + (-S))(x) = S(x) + (-S)(x) = S(x) - S(x) = 0 \end{align}
(6)
\begin{align} \quad ((-S) + S)(x) = (-S)(x) + S(x) = -S(x) + S(x) = 0 \end{align}
  • Associativity of Scalar Multiplication:
(7)
\begin{align} \quad \alpha (\beta S(x)) = \alpha \beta S(x) = (\alpha \beta) S(x) \end{align}
  • Existence of a Multiplicative Identity: For $1 \in \mathbb{C}$ we have that:
(8)
\begin{align} \quad 1 \cdot (S(x)) = S(x) = S(x) \cdot 1 \end{align}
  • Distributivity:
(9)
\begin{align} \quad \alpha (S + T)(x) = \alpha (S(x) + T(x)) = \alpha S(x) + \alpha T(x) \end{align}
(10)
\begin{align} \quad (\alpha + \beta)S(x) = \alpha S(x) + \beta S(x) \end{align}
  • Therefore $\mathcal L(X, Y)$ is a vector space with these operations. $\blacksquare$
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