Linear Operators on Linear Spaces

Linear Operators on Linear Spaces

Definition: Let $X$ and $Y$ be linear spaces over $\mathbb{R}$ (or $\mathbb{C}$). A function $T : X \to Y$ is said to be a Linear Operator from $X$ to $Y$ if it satisfies the following properties:
1) $T(x + y) = T(x) + T(y)$ for all $x, y \in X$. (Additivity)
2) $T(\lambda x) = \lambda T(x)$ for all $x \in X$ and for all $\lambda \in \mathbb{R}$ (or $\mathbb{C}$). (Homogeneity)
The set of all linear operators from $X$ to $Y$ is denoted $\mathcal L (X, Y)$.

It is common to use the notation "$Tx$" instead of "$T(x)$" to denote the image of $x$ under $T$.

For example, let $X = \mathbb{R}$ and $Y = \mathbb{R}$ and consider the function $T : \mathbb{R} \to \mathbb{R}$ defined for all $x \in \mathbb{R}$ by:

\begin{align} \quad T(x) = 2x \end{align}

Then $T$ is a linear operator from $\mathbb{R}$ to $\mathbb{R}$ since for all $x, y \in \mathbb{R}$ we have that:

\begin{align} \quad T(x + y) = 2(x + y) = 2x + 2y = T(x) + T(y) \end{align}

And for all $x \in \mathbb{R}$ and for all $\lambda \in \mathbb{R}$ we have that:

\begin{align} \quad T(\lambda x) = 2(\lambda x) = \lambda (2x) = \lambda T(x) \end{align}
Definition: Let $X$ and $Y$ be normed linear spaces over $\mathbb{R}$ (or $\mathbb{C})$. A linear operator $T : X \to Y$ is said to be a Bounded Linear Operator if there exists an $M \in \mathbb{R}$, $M \geq 0$ such that for every $x \in X$ we have that $\| T(x) \| \leq M \| x \|$. The set of all bounded linear operators from $X$ to $Y$ is denoted $\mathcal B(X, Y)$.

Note that we can only consider bounded linear operators if both $X$ and $Y$ are normed linear spaces.

From the example above, we see that with the standard Euclidean norm of the absolute value on $X = Y = \mathbb{R}$, that $T : \mathbb{R} \to \mathbb{R}$ defined for all $x \in \mathbb{R}$ by $T(x) = 2x$ is a bounded linear operator since for all $x \in \mathbb{R}$:

\begin{align} \quad \| T(x) \| = |T(x)| = |2x| = 2|x| = 2\|x \| \end{align}

Where $M = 2$.

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