# Vectors

In linear algebra, you learned about vectors, and you saw that
the properties of vectors may be summarized as follows:

A collection **V** of object **v**
is called a *real vector space* if the two operators + and ⋅ (called *vector
addition* and *scalar multiplication*) are defined and
satsify:

- If
**u** ∈ **V** and **u** ∈ **V** then **u** + **v** ∈ **V**.
**u** + **v** = **v** + **u** ∀ **u**, **v** ∈ **V** (commutativity).
- (
**u** + **v**) + **w** = **u** + (**v** + **w**) ∀ **u**, **v**, **w** ∈ **V** (associativity).
- ∃
**0** ∈ **V** such that **0** + **v** = **v** ∀ **v** ∈ **V** (the zero vector).
- ∀
**v** ∈ **V**, ∃ **u** ∈ **V** such that **v** + **u** = **0**.
*a*⋅(*b*⋅**v**) = (*a**b*)⋅**v** ∀ **v** ∈ **V** and *a*, *b* ∈ **R**.
- 1⋅
**v** = **v** where 1 ∈ **R**.
*a*⋅(**u** + **v**) = *a*⋅**u** + *a*⋅**v** ∀ **u**, **v** ∈ **V** and *a* ∈ **R**.
- (
*a* + *b*)⋅**v** = *a*⋅**v** + *b*⋅**v** ∀ **v** ∈ **V** and *a*, *b* ∈ **R**.

If you try this out with all 3-dimensional vectors (*a*, *b*, *c*)^{T}, you will
find that all of these properties hold.

# Functions

If you consider real-valued functions f(*x*), g(*x*), and h(*x*) of a variable *x*, and try all of the above
requirements, you will find that all of the properties hold. Therefore, the collection of all real-valued
functions of a variable form a vector space. The zero vector is the zero function.

# Dot Products

The dot product is the sum of the pairwise products of two vectors. The corresponding
result for functions is the integral of the products:

Now, we define the *2-norm* of a vector as being the square root of the dot product of
the vector with itself:

Thus, we can define the 2-norm of a function as:

Note, however, that not all functions have a finite 2-norm. For example, if we are integrating
from −∞ to ∞, the norm of the function f(*x*) = 1 is ∞. Thus, we have
to restrict ourselves to functions which have a finite 2-norm. Fortunately, the collection of all
functions which satisfy this condition is still a vector space.

# Other Norms

We can define other norms, similar to the 1-norm and the ∞-norm of vectors:

The *sup*, or supremum, of a function is a generalization of the
maximum of a function.

# So, What is a Matrix for Functions?

If functions are vectors, can't we have matrices, too? The answer is: *yes*, and you've
already seen them (or will see them). Remember that a matrix is nothing more than a bunch
of row vectors, and the result of a matrix-vector product is the dot-product of each of
the row vectors with the vector. If we define the *i*th row of **M** by
**M**_{i}, then we may write a matrix-vector product as:

To extend this to functions, let us consider only those function which are defined on the interval [0, &infin).
Define a function of two variables **L**(*x*,*y*) = e^{-xy}.
This function is symmetric in *x* and *y*. Thus, let us define the *product* of this
function **L** with a function f(*x*) as

You have already seen this *matrix-for-functions*: it is the Laplace transform. Instead of calling
the function **L** a matrix, it is called an *operator*.

Now you should be thinking: if we can define norms on matrices, why can we not define a norm on an
operator? That's for later on...

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.