Unifying Different Definitions of Adjoint Map

[WWGD]

Hi, this question seem to fall somewhere between Analysis and Algebra;
I just choose this section; sorry if it is the wrong one. I would appreciate
any suggestions, refs., etc.

I'm basically trying to see if the different definitions of adjoint
maps can be unified into a single definition:

1) Adjoint of a linear map: given a linear map L:V-->W between f.d
vector spaces V,W , the adjoint map L*: W*-->V* is given by, for w* in W*,
for v in V :

L*[(w*)(v)]:=w*(L(v)).

2) Given an inner-product space (V, < ,>) , and a linear map L:V-->V,
the adjoint L* (which may not exist if V is infinite-dimensional) is
defined as L* satisfying:

<Lv,w>=<v,L*w>

for any v,w in V.

Can these two definitions of adjoint be unified into a single, broader
definition of adjoint map?

 

[mathman]

They are basically the same. The first definition is more general. The second is specifically for inner product spaces.

 

[WWGD]

Thanks, but, in what sense are they basically the same? Can you think of an inner-product so that the linear dual can be expressed as an adjoint and viceversa, can the inner-product adjoint be expressed as a linear dual somehow?

 

[micromass]

Thanks, but, in what sense are they basically the same? Can you think of an inner-product so that the linear dual can be expressed as an adjoint and viceversa, can the inner-product adjoint be expressed as a linear dual somehow?

The adjoint from (1) is usually called the Banach adjoint. The adjoint in (2) is the Hilbert adjoint. The two are not special cases of some general formula. In particular, (2) is not a special case of (1).

Let's work in general Banach and Hilbert spaces here, but let's take everything finite-dimensional (extensions to infinite dimensions exist, but are more technical). So a Banach space here will be a finite-dimensional real vector space (this can always be equipped with a suitable and essentially unique norm). A Hilbert space here is a finite-dimensional real inner-product space.

However, we can use (1) to define (2). The reason is that a Hilbert space satisfies a very particular property called the Riesz representation theorem. It says that

\Phi<img src="/styles/physicsforums/xenforo/smilies/arghh.png" class="smilie" loading="lazy" alt=":H" title="Gah! :H" data-shortname=":H" />\rightarrow H^*:x\rightarrow &lt; -,x&gt;

is a isomorphism of vector spaces for each Hilbert space.

Then let ##L_1\rightarrow H_2$be a linear map between Hilbert spaces. Let$\Phi_1$and$\Phi_2$be the above maps for$H_1$and$H_2$respectively. Let$L^\prime_2^*\rightarrow H_1^*$be the Banach adjoint. Then we can define the Hilbert adjoint as$L^*(x) = \Phi_1^{-1}(L^\prime(\Phi_2(x)))$. It is sometimes standard to identify$H$with$H^\prime$through$\Phi$. So we set$x = <-,x>##. With this abuse of notation, the Hilbert adjoint is a special case of the Banach adjoint.

 

[WWGD]

Yes, I understand that, I know of the Riesz representation theorem, and I too am aware of Riesz representation and how it is used to define the Hilbert adjoint, but the question is whether one of these is a form of the other, or if there is a more general definition that covers both. Maybe I didn't ask the question clearly-enough.

 

[HallsofIvy]

Given a vector space, V, its dual, V*, is the space of linear "functionals" on V- that is, the set of a linear functions from V to the real numbers. Given an "inner product space", with inner product, <u, v>, we can associate the functional f(x)= <x, v> to the vector v. That is an isomorphism from the inner product space V to its dual, V*, and maps the "adjoint" defined on inner product spaces to the "adjoint" defined on the general vector spaces.

(That's essentially what micromass said. I don't see why you do not think he completely answered your question.)

 

[WWGD]

Here's another answer in case anyone's interested:

http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst&task=show_msg&msg=4631.0001

 

[mathwonk]

here is my attempt to explain adjoints, see pages 50-60:

http://www.math.uga.edu/%7Eroy/4050sum08.pdf