Understanding the Complex Inner Product: Definition and Importance

[unchained1978]

When studying complex numbers/vectors/functions and so forth you constantly encounter the idea of an inner product of two quantities (numbers/vectors/functions). It's represented as A*B is the inner product of two of these, but I've never been convinced why it couldn't also be AB*, as this is some cases yields a different answer. It seems arbitrary to me that it is defined in either way. Can someone explain to me the correct definition and also why it is correct? Any help would be much appreciated.

 

[Fredrik]

Most math books define inner products and semi-inner products to be linear in the first variable and antilinear in the second. Most physics books define them to be linear in the second variable and antilinear in the first. These are just two different conventions.

What do you mean by A*B and AB*. Are A and B complex numbers, n×1 matrices, or something else? If they are n×1 matrices, then <A,B>=AB* doesn't work, since the right-hand side is an n×n matrix. (Note that A* and B* are 1×n matrices). But you could define <A,B>=B*A if you want to.

 

[unchained1978]

Just talking about vectors and functions for the moment. The star denotes the complex conjugate of the element to the left of it in the way I've written it. The motivation for the question is just that, if you consider two vectors A and B, then A*B=/AB* for the general case, and vice versa. I'm trying to understand if it's merely a convention to define it this way or if there exists some mathematical reasoning behind it, because the fact that two different products arise from the same two vectors seems to be a problem to me.

 

[Fredrik]

The inner product on ℂ is pretty useless, since ℂ is a 1-dimensional vector space.

If the vector space is the set of functions from a set X into ℂ, then <f,g>=f*g isn't an inner product, since f*g is a function, not a member of ℂ. (The "complex conjugate" of a function is defined by f*(x)=f(x)* for all x, and the product of two functions is defined by (fg)(x)=f(x)g(x) for all x, so f*g is the function that takes x to f(x)*g(x)).