Notations for dual spaces and dual operators

Question

I'm asking for opinions about the 'best' notations for: 1. the algebraic dual of a vector space $X$; 2. the continuous dual of a TVS; 3. the algebraic dual (transpose) of an operator $T$ between vector spaces; 4. the dual (transpose) of a continuous operator between TVS; 5. the adjoint of a bounded operator $T$ between Hilbert spaces.

My problem is that I would like to use these notions in the same context. The standard notations tend to overlap but I am forced to use different notations for each of these entities. Of course it is easy to come up with notations, but some traditions are well established and it is not trivial to respect them and at the same time keep them apart, with some elegance.

What I'm using now: 1. $X'_{alg}$ 2. $X'$ 3. $^tT$ 4. $T'$ 5. $T^*$

Thank you for your advice.

Answer 1

The standard notation don't overlap as much as you think if you are careful with types. For example $X'$ is perfectly fine for both 1. and 2. because vector spaces and topological vector spaces are not the same type. Any object $X$ should only have one type so that there can never be any confusion. You can use the same notation if you never forget forgetful functors (i.e. never pretend that a TVS is the same as its underlying vector space). Similarly ' can be used for both 3. and 4.

If you do want to extract the underlying vector space of a TVS some times, you might have an easier time just giving a name to the forgetful functor instead of inventing more notation.

EDIT:

And by the way: You don't have to name the forgetful functor $F$ or something similar like Nate used in the comment. You can also name it implicitely, although that solves only half of your problems. Simply stop using the abuse-of-notation $X$ when you mean $(X,\tau)$ for a vector space topology $\tau$ (which we already knew could happen when we introduced this and any other abuse-of-notation. They live only as long as they're useful and don't lead to confusion!) and the functor becomes $(X,\tau)\mapsto X$. Then you can use both $(X,\tau)'$ and $X'$ without confusion. Of course problems 3. and 4. persist.

Answer 2

Given your situation of having to juggle all these notational traditions at the same time, I would recommend for a space $X$ and an operator $A$:

1. $X^{\vee}$, 2. $X'$, 3. $A^{\rm T}$, 4. $A^{\rm T}$, 5. $A^*$

My rationale is as follows.

For 1: Algebraic geometers especially use the "vee' notation for the algebraic dual so I think the cultural association helps the brain automatically make the association with the algebraic notion of dual.

For 2: One usually does not write the space of tempered distributions as $S^*(\mathbb{R}^d)$ but rather as $S'(\mathbb{R^d})$.

For 3 and 4: I would use the same notation as per the comment by user95282 since the maps are related by restriction. One could use ${}^{\rm t}A$ for both but I prefer the matrix algebra notation if only because it is easier to type.

For 5: As yuggib said, it is standard in the theory of $C^*$-algebras and spectral theory. I don't see any reason to be a contrarian and not follow what everyone else does.

Answer 3

I like the notation $T^*$ for the adjoint, since it agrees with the usual notation for the involution of *-algebras. I don't know however if I would distinguish between $^t T$ and its continuous version with an ad hoc notation.

For algebraic and continuous duals of (topological) vector spaces, personally I like Bourbaki's notation $X^*$ for the algebraic dual and $X'$ for the continuous dual (even if $X_{\prime}$ for preduals is afwul, as opposed to $X_*$ that is natural if $X^*$ is the continuous dual). In addition, if $X$ is also a (subset of some) *-algebra, the notation $X^*$ may yield some confusion.