Definition:Pseudoinverse of Bounded Linear Transformation
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Definition
Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.
Let $S: X \to Y$ be a bounded linear transformation.
Let $T: Y \to X$ be a bounded linear transformation.
$S$ and $T$ are pseudoinverse to each other if and only if:
- $T \circ S - I_X$ is compact
and:
- $S \circ T - I_Y$ is compact
where:
- $\circ$ denotes the composition
- $I_X$ denotes the identity mapping of $X$
- $I_Y$ denotes the identity mapping of $Y$
This article, or a section of it, needs explaining. In particular: $T \circ S - I_X$ and $S \circ T - I_Y$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
This article is complete as far as it goes, but it could do with expansion. In particular: Establish the conceptual connection between compactness and degeneracy, as the definition is different according to whether L.T. is bounded or not You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also see
Sources
- 2002: Peter D. Lax: Functional Analysis: $27.1$: The Noether Index