Definition talk:Adjoint (Norm Theory)

There is something amiss here, as the definition uses inner products, which are not defined for a normed vector space. Also, we cannot have both $x, y \in Y$.

Referring to my source ( Book:Reinhold Meise/Introduction to Functional Analysis ), I can see two possibilities:

• either $X, Y$ are assumed to be Hilbert spaces, and $A$ is bounded, in which case the definition is equal to Definition:Adjoint Linear Transformation with $x \in X, y \in Y$.

• or there is an omission, such that $A^*$ is defined as $A^* : Y^* \to X^*$, where $X^*, Y^*$ are the normed dual spaces of $X, Y$. In this case, we should have $x \in X, y \in Y^*$, and the notation $\innerprod {A x} y$ should be interpreted as $\map y {Ax}$. Then $A^*$ is what Meise calls an adjoint map.

Maybe you can take another look in your source? --Anghel (talk) 18:08, 12 December 2022 (UTC)

"1 b. also called dual, the operator $A^*$ that is conjugate to a given linear operator $A$ between normed spaces $X$ and $Y$; it is defined by $\innerprod {A x} y = \innerprod x {A^* y}$ where $\innerprod {\,} {\,}$ represents a pairing between a space and its dual, and $A^*$ maps $Y^*$ into $X^*$. (See also sense 4.)"

Sense 4 says: "Hilbert space adjoint, the operator $A^*$ that is conjugate to a given linear operator $A$; it is defined on a hilbert space by $\innerprod {A x} y = \innerprod x {A^* y}$. In this case $\paren {c A}^* = \overline c A^*$, while if $A^*$ is viewed as a mapping between dual spaces, as in sense 1 b, then $\paren {c A}^* = c A^*$."

I missed out the $*$ against $X$ and $Y$. --prime mover (talk) 20:31, 12 December 2022 (UTC)