Definition:Invertible Bounded Linear Transformation/Normed Vector Space
Definition
Let $\struct {V, \norm \cdot_V}$ and $\struct {U, \norm \cdot_U}$ be normed vector spaces.
Let $A : V \to U$ be a bounded linear transformation.
We say that $A$ is invertible as a bounded linear transformation if and only if:
- $A$ has an inverse mapping that is a bounded linear transformation.
That is:
- there exists a bounded linear transformation $B : U \to V$ such that:
- $A \circ B = I_U$
- $B \circ A = I_V$
where $I_U$ and $I_V$ are the identity mappings on $U$ and $V$ respectively.
We say that $B$ is the inverse of $A$ and write $B = A^{-1}$.
The process of finding an $A^{-1}$ given $A$ is called inverting.
Notes
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Note that a mapping is usually called invertible if it has an inverse mapping.
However, in the context of a bounded linear transformation, it often makes sense to insist that this inverse mapping be bounded.
Some sources therefore call a bounded linear transformation $A$ with bounded inverse invertible, while simply saying that $A$ has an inverse otherwise.
To avoid confusion, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we emphasise that $A$ is invertible in the sense of a bounded linear transformation or invertible as a bounded linear transformation, if a bounded inverse is insisted.
Where boundedness of the inverse is not required, we can simply call $A$ invertible, invertible as a mapping or invertible in the sense of a mapping.