Talk:Complex conjugate of a vector space

Some ideas which came up when I read the article:

In some applications that I have seen, one needs to consider both a vector space V, its dual space, its complex conjugate space, and even the dual of the complex conjugate space. In this case, a notation which distinguishes the dual space from the complex conjugate space is needed. In the current article, the same notation is used for the complex conjugate space (a star sign) as is used for the dual space in its article. Suggestion: either change the notation to distinguish it from the dual space, or explain that sometimes different notations are needed.

I lack a good motivation for the introduction of the complex conjugate space. I am not an expert in this field myself and have no suggestion, but I guess that there are some in physics?

I get the feeling that the current article is a mix between basic facts, such as how a complex conjugate space is constructed, and advanced material such as functors and categories. I doubt that many who reads about complex conjugate spaces for the first time here have any idea what functors and categories are. I will try to make a separation of the two types of information.

--KYN 00:11, 31 December 2005 (UTC)[reply]

I agree whole-heartedly with the first point. On overline for complex conjugation would be better for that reason.

Shambolic Entity 02:24, 3 November 2006 (UTC)[reply]

I changed all the stars to overlines. Since I don't know how to make overlines on normalsize letters, all math symbols are now in larger font, hope that's OK. (The stars didn't look very good, either.) I added one sentence of motivation.--345Kai (talk) 09:37, 21 September 2008 (UTC)[reply]

1. The concept of the conjugate space and its dual have puzzled me for a long time. I have attempted several definitions, none of which I have found to be consistent.

2. The definition of the conjugate space given in this article does not make sense to me. To be able to form the complex conjugate of a vector in an unique way, one needs first to specify a basis - or rather, one needs to specify what conjugation does to a basis - before the operation becomes unique. This seems to require an additional structure over and above the complex linear structure of the complex vector space. I have not found a formalisation of this additional structure that satisfies me.

3. The concept of conjugate spaces and of dual conjugate spaces does arise in physics - in particular in QM - but it arose in mathematics long before the modern quantum theory, in the theory of the representation of groups. It occurs in particular in the work of Isiah Schur. So, for example, GL(2,C) has four basic representation from which all others are constructed, the vector space, the dual space, the complex conjugate, and the dual complex conjugate. The subgroup SU(2,C) has only two distinct basic representations, the other two being equivalent to it. Complex representation theory is probably the place to look. One usually demands of representations that they be unitary, in which case there is an associated Hermitian form on the space, but this is bu no means necessary. For example, the Dirac theory uses non-unitary representations (it has to, since the Lorentz group is not compact, and all unitary representations of non-compact groups are infinite dimensional, whereas the Dirac theory works with 4-dimensional representations).

4. A recent idea that has vexed me is that the complex conjugate space may not be a distinct space at all, and that what we may need to define complex conjugate vectors is an antilinear endomorphism on the space itself. But this too requires some additional structure to give it meaning. TulliusAgrippa (talk) 08:35, 22 June 2009 (UTC)[reply]

Complex conjugate vector space vs.Dual vector space

This topic definitely needs to be contrasted with Dual vector space so that the distinction is clear. For instance, the last section states that in quantum mechanics, the conjugate to a ket vector is a bra vector, but of course bras are the duals to kets (as it confirms in the article on bra-ket notation. I assume $V\;$ , ${\overline {V}}$ , $V^{*}\;$ , and ${\overline {V}}^{*}$ are all isomorphic to each other, and the only difference between $V^{*}\;$ and ${\overline {V}}$ is the nature of the isomorphism. In particular, I think the dual $V^{*}\;$ is in some sense a "different" vector space than $V\;$ (the space of scalar functions), while the complex conjugate ${\overline {V}}$ is literally the same set of vectors $V\;$ , only with $V=(v_{1},v_{2},...)\;$ identified with ${\overline {V}}=({\overline {v}}_{1},{\overline {v}}_{2}^{*},...)\;$ , where ${\overline {v}}\;$ denotes the scalar complex conjugate of $v\;$ .

Could someone more knowledgeable than me make this precise? 69.254.150.148 (talk) 13:37, 6 April 2011 (UTC)[reply]

Mistake?

Let's looks at the definition of product by a scalar: $\alpha \cdot {\overline {v}}:={\overline {\,{\overline {\alpha }}\,v\,}}$

Assuming $v$ can be written as $(a+ib){\vec {w}}$ (complex number times a "real" vector, or maybe a sum of such terms, it doesn't change anything...), one has

{\overline {\,{\overline {\alpha }}\,v\,}}={\overline {{\overline {\alpha }}(a+ib){\vec {w}}}}={\overline {\overline {\alpha }}}(a-ib){\vec {w}}=\alpha {\overline {v}}

it's the same multiplication as in V. What's the point that?

Wheareas in the french version, one has a much clearer definition

\alpha *{\vec {v}}:={\overline {\alpha }}\cdot {\vec {v}}

where $\cdot$ is the scalar product of the original V, while the equality is the definition of $*$ Noix07 (talk) 14:34, 16 March 2015 (UTC)[reply]

I agree. The French version must either be correct or the section on conjugate linear maps is incorrect. Notice that defining the map as

{\bar {f}}({\bar {v}})={\overline {f(v)}}

will preserve (anti)-linearity of f, rather than changing between linear/antilinear. Kreizhn (talk) 18:44, 20 March 2015 (UTC)[reply]

Actually I've done the math: For the map

V\to {\bar {W}}

taking

{\bar {v}}\mapsto Tv

to take antilinear to linear, it must be the case that

\alpha *{\bar {v}}=\alpha {\bar {v}}

.

As an aside, I think that regardless of the definition of scalar multiplication, the map

{\bar {f}}{\bar {v}}={\overline {fv}}

must always preserve (anti)-linearity. If the complex conjugation map

C:V\to {\bar {V}}

interchanges linearity with antilinearity, then the given map is just

{\bar {f}}=C_{W}\circ f\circ C_{V}:{\bar {V}}\to {\bar {W}}

. Hence (anti)linearity is always preserved. Kreizhn (talk) 18:59, 20 March 2015 (UTC)[reply]

I suggest that we get rid of the redundant overbars on vectors, which don't serve any purpose and confuse the hell out of any reader. (If I understanding the wording of the article, the overbarred

{\overline {v}}

is just a copy of

v

in the conjugate space.) I can't access the cited reference. So I don't know what is said there, but here is a note that explains the issues quite clearly [1]. An authentic source here.^[1] - SindHind (talk) 23:01, 26 August 2015 (UTC)[reply]

Done I have done this cleanup now. - SindHind (talk) 15:37, 27 August 2015 (UTC)[reply]

References

^ K. Schmüdgen (11 November 2013). Unbounded Operator Algebras and Representation Theory. Birkhäuser. pp. 16–. ISBN 978-3-0348-7469-4.

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1. The concept of the conjugate space and its dual have puzzled me for a long time. I have attempted several definitions, none of which I have found to be consistent.

2. The definition of the conjugate space given in this article does not make sense to me. To be able to form the complex conjugate of a vector in an unique way, one needs first to specify a basis - or rather, one needs to specify what conjugation does to a basis - before the operation becomes unique. This seems to require an additional structure over and above the complex linear structure of the complex vector space. I have not found a formalisation of this additional structure that satisfies me.

3. The concept of conjugate spaces and of dual conjugate spaces does arise in physics - in particular in QM - but it arose in mathematics long before the modern quantum theory, in the theory of the representation of groups. It occurs in particular in the work of Isiah Schur. So, for example, GL(2,C) has four basic representation from which all others are constructed, the vector space, the dual space, the complex conjugate, and the dual complex conjugate. The subgroup SU(2,C) has only two distinct basic representations, the other two being equivalent to it. Complex representation theory is probably the place to look. One usually demands of representations that they be unitary, in which case there is an associated Hermitian form on the space, but this is bu no means necessary. For example, the Dirac theory uses non-unitary representations (it has to, since the Lorentz group is not compact, and all unitary representations of non-compact groups are infinite dimensional, whereas the Dirac theory works with 4-dimensional representations).

4. A recent idea that has vexed me is that the complex conjugate space may not be a distinct space at all, and that what we may need to define complex conjugate vectors is an antilinear endomorphism on the space itself. But this too requires some additional structure to give it meaning.

Last edited at 08:23, 22 June 2009 (UTC). Substituted at 01:54, 5 May 2016 (UTC)

[Schmüdgen2013-1] K. Schmüdgen (11 November 2013). Unbounded Operator Algebras and Representation Theory. Birkhäuser. pp. 16–. ISBN 978-3-0348-7469-4.

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