Inner Product Spaces

[Prometheus]

Just trying to clarify my understanding of some concepts regarding inner products and spaces.

 

So the inner product in Dirac notation has been defined in my book as:

 

[math]\langle a|b\rangle = \sum_i a_i^*b_i[/math]

 

Does this definition include the complex conjugate, [math]a_i^*[/math], just so that the property

[math] \langle a|a\rangle = \sum_i |a_i|^2[/math] is satisfied?

 

i.e. is that property a defining feature or the corollary of some other defining feature?

 

 

Also, is an inner product space then just the space spanned by the two vectors that form the inner product?

 

For instance the inner product of cubic polynomials has been defined in my book as:

 

[math]\langle a|b\rangle = \int_{-1}^1a^*(x)b(x)dx[/math]

 

but i think we could also define it as [math] \langle a|b\rangle = \int_{-c}^ca^*(x)b(x)dx [/math] for any real (and complex?) c.

 

If so would we have defined a different inner product space to the first one, or are they somehow the same?

[Mordred]

I'm way out of practice on Dirac lol Let me think about this. Its been awhile since I practiced Bra and Ket etc.

 

Just out of curiousity which textbook do you have. I may have a copy of it.

 

If I recall correctly part 1 is correct. But need to double check. Still thinking on the complex conjugate.

 

PS I hope you won't mind if I reply in a manner that help others understand this notation. Subject rarely comes up

Edited December 22, 2016 by Mordred

[studiot]

We want our inner product to be a map from a vector space to the Real numbers ie a scalar.

 

Consider any complex number a+ib

 

(a + ib)² = (a + ib) (a + ib) = a² + 2iab - b²

 

Which is another complex number ie another vector.

 

To get a real number we must multiply by the complex conjugate

 

(a + ib) (a - ib) = a² + iab - iab + b² = a² + b²

 

This will always be a positive number we can take the real square root of.

 

Don't forget that vectors here are Euclidian vectors and the inner product represents the Euclidian norm with is defined as the square root of the sum of the products of the coordinates.

Edited December 22, 2016 by studiot

[Prometheus]

I'm way out of practice on Dirac lol Let me think about this. Its been awhile since I practiced Bra and Ket etc.

 

Just out of curiousity which textbook do you have. I may have a copy of it.

 

If I recall correctly part 1 is correct. But need to double check. Still thinking on the complex conjugate.

 

PS I hope you won't mind if I reply in a manner that help others understand this notation. Subject rarely comes up

 

I doubt it, i'm following an open uni book. I can pm a pdf of it if you like.

 

Of course i don't mind, it'll likely help me too.

 

 

We want our inner product to be a map from a vector space to the Real numbers ie a scalar.

 

Consider any complex number a+ib

 

(a + ib)² = (a + ib) (a + ib) = a² + 2iab - b²

 

Which is another complex number ie another vector.

 

To get a real number we must multiply by the complex conjugate

 

(a + ib) (a - ib) = a² + iab - iab + b² = a² + b²

 

This will always be a positive number we can take the real square root of.

 

Don't forget that vectors here are Euclidian vectors and the inner product represents the Euclidian norm with is defined as the square root of the sum of the products of the coordinates.

 

Happy all the way up to the inner product being the Euclidean norm. I can follow the algebra from the definitions easy enough, but i can't internalise it. So these vectors are representations of functions (are they still classed as Euclidean vectors then?): i just can't picture what it means for a function to have a magnitude. Maybe it's my understanding of norms that needs addressing.

[mathematic]

Inner product spaces are best viewed as mathematical abstractions. They are vector spaces (finite or infinite dimension) over a scalar field (real or complex numbers) where an inner product is defined obeying certain relationships. A norm (distance function) is defined by the square root of the inner product of a vector and itself (real) or with its complex conjugate (complex).

 

If the space is complete (contains all its limit points) then finite dimensional space is called Euclidean while infinite dimensional space is call Hilbert.

[Prometheus]

So basically don't try to picture it?

[mathematic]

So basically don't try to picture it?

Why not? For 2 and 3 dimensional spaces, we use ordinary Euclidean geometry.

[Prometheus]

Why not?

 

Because it hurts my head. This may be because i have little formal maths education, so let me dumb this down a bit. When i imagine a function, i picture some squiggly line. When i imagine a vector i imagine a straight line, ending at some point. Both of these occur in a coordinate system with real numbers. The first thing my brain tries to do when imagining that one function is orthogonal to another is to have one squiggly line go off somewhere and another squiggly line at a sort of right angle to the first. But that's not what is going on here.

 

I think i may be more comfortable just following the definitions without trying to imagine what is happening.