Linear Operators and Bounded Linear Operators Examples 1

We will now look at some example problems regarding linear operators and bounded linear operators.

Example 1

Proposition 1: Let $(X, \| \cdot \|_X)$ and $(Y, \| \cdot \|_Y)$ be normed linear spaces. If $(T_n) \subseteq \mathcal B(X, Y)$ converges to $T \in \mathcal B(X, Y)$ and $(x_n)$ converges to $x \in X$ then $(T_n(x_n))$ converges to $T(x) \in Y$.

• Proof: Let $\epsilon > 0$ be given. Since $(T_n)$ converges to $T$ there exists an $N_1 \in \mathbb{N}$ such that if $n \geq N_1$ then:

(1)

\begin{align} \quad \| T_n - T \| < \epsilon \quad (*) \end{align}

• Since $(x_n)$ converges to $x$ there exists an $N_2 \in \mathbb{N}$ such that if $n \geq N_2$ then:

(2)

\begin{align} \quad \| x_n - x \|_X < \epsilon \quad (**) \end{align}

• Also note that since $(x_n)$ converge to $x$ we have that $(\| x_n \|)$ is bounded. (To see this, if $\epsilon = 1$ for example, then for some $N \in \mathbb{N}$ if $n \geq N$ we have that $\| x_n - x \|_X < 1$. So $\| x \|_X < 1 + \| x \|_X$. If $M = \max \{ \| x_1 \|_X, \| x_2 \|_X, ..., \| x_{N-1} \|_X, 1 + \| x \|_X \}$ then $\| x_n\|_X \leq M$ for all $n \in \mathbb{N}$.) So there exists some $M > 0$ for which $\| x_n \| \leq M$ for all $n \in \mathbb{N}$.

• Let $N = \max \{ N_1, N_2 \}$. Then if $n \geq N$ we have that both $(*)$ and $(**)$ hold and so:

(3)

\begin{align} \quad \quad \| T_n(x_n) - T(x) \|_Y = \| T_n(x_n) - T(x_n) + T(x_n) - T(x) \|_Y \leq \| T_n(x_n) - T(x_n) \|_Y + \| T(x_n) - T(x) \| \leq \| T_n - T \| \| x_n \|_X + \| T \| \|x_n - x \|_X < \epsilon [M + \| T \|] \end{align}

• So $(T_n(x_n))$ converges to $T(x) \in Y$. $\blacksquare$