# Category:Adjoints

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This category contains results about **adjoint linear transformations** in the context of **Hilbert Spaces**.

Let $\HH$ and $\KK$ be Hilbert spaces.

Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.

Let $A \in \map \BB {\HH, \KK}$ be a bounded linear transformation.

By Existence and Uniqueness of Adjoint, there exists a unique bounded linear transformation $A^* \in \map \BB {\KK, \HH}$ such that:

- $\forall h \in \HH, k \in \KK: {\innerprod {\map A h} k}_\KK = {\innerprod h {\map {A^*} k} }_\HH$

where $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ are inner products on $\HH$ and $\KK$ respectively.

$A^*$ is called the **adjoint** of $A$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### D

### E

### H

## Pages in category "Adjoints"

The following 15 pages are in this category, out of 15 total.