Definition:Linear Isomorphism

Definition

Let $\GF \in \set {\R, \C}$.

Let $X$ and $Y$ be normed vector spaces over $\GF$.

Let $T : X \to Y$ be a bijective bounded linear transformation.

We say that $T$ is a linear isomorphism if and only if it is invertible as a bounded linear transformation.

That is, a bijective linear transformation is a linear isomorphism if and only if it is bounded with bounded inverse.

If there exists a linear isomorphism $T : X \to Y$, we say that $X$ and $Y$ are (linearly) isomorphic.

Also see

• Definition:Isometric Isomorphism - a stronger concept that additionally requires $T$ to preserve norm

Sources

• 2001: Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant and Václav Zizler: Functional Analysis and Infinite-Dimensional Geometry ... (previous) ... (next): Proposition $1.19$