Review of Linear Operators

• Recall from the Linear Operators Between Vector Spaces page that if $E$ and $F$ are vector spaces then a function $f : E \to F$ is a Linear Operator from $E$ to $F$ if for all $x, y \in E$ and for all $\lambda \in \mathbf{F}$ we have that $f(x + y) = f(x) + f(y)$ and $f(\lambda x) = \lambda f(x)$.

• When $E$ and $F$ are topological vector space, we have the following equivalences for when $f : E \to F$ is continuous.

	Criteria for Continuity in a Topological Vector Space
(1)	$f$ is continuous.
(2)	$f$ is continuous at the origin.
(3)	There exists a constant $M > 0$ such that $\| f(x) \| \leq M \| x \|$ for all $x \in E$. (This requires that $E$ and $F$ are normed spaces).

• On the Isomorphic Topological Vector Spaces we said that two topological vector spaces $E$ and $F$ are Isomorphic if there exists a bijective linear operator $f : E \to F$ for which both $f$ and $f^{-1}$ are continous.

• We then proved that if $E$ and $F$ are normed spaces and $f : E \to F$ is a linear bijection, then $f$ is an isomorphism if and only if there exists constants $M, N > 0$ such that for all $x \in E$:

(1)

\begin{align} \quad M \| x \| \leq \| f(x) \| \leq N \| x \| \end{align}