Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

what is an Inner Product Space?

taken from my book:

Let V be a real vector space. Supppose to each pair of vectors u,v ∈  V there is assigned a real number, denoted by <u,v>. This function is called a (real) inner product on V if it satisfied the following axioms:

linear property: <au1 + bu2, v> = a<u1, v> + b<u2,v>
symmetric property: <u,v> = <v,u>
positive definite property: <u,u> >= 0; and <u,u> = 0 if and only if u = 0.

The vector space V with an inner product is called a (real) inner product space.

everything but the last line makes sense. What do they mean a vector space 'with' an inner product? Does this function create some sort of vector space itself? That is, is an inner product space a vector space? If so, what do the elements consist of?

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A logarithm is just a misspelled algorithm.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: what is an Inner Product Space?

Well, the "space" concept comes from the desire of mathematicians to do addition, substraction, multiplication, division ... freely. "Freely" means if they do such an algebra of elements in the space, they get the result from this space too. That's why they invented negative numbers and radicals and imaginary numbers-they don't wanna tell you I am stuck in telling you the answer of 2-4 or that of sqrt(3) or x^2=-2.

So a space in algebra term is that you play certain manuplications in it and you are sure the results are also in it. Integers form a good space for addition and substraction, well natuaral numbers don't. Vectors with the same degree can also form a good space for addition and substraction, and for scale product k v=  (kv1, kv2, kv3), They call it a linear space because combinately you are sure the result of a u + b v is in the vector space too.  Trivally, {(0,0,0)} is also a linear space for it satisfy the definition of a linear space.

Now they want more than a linear space. They want each two vectors from an already linear space to have a fixed number as their so called inner product, additionally a positive number if the two are the same, 0 when one is 0 vector. It does not matter how the inner product is defined, as long as the defined inner product can be applied to any two and satisfy those conditions. They call this linear space also an inner product space.

Q: Is there any inner product space not linear space?
A: No, they are defined with linear property. In other words, mathematicians don't think it is useful to define a space satisfying inner product property but not  linearity.

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X'(y-Xβ)=0