# 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:

1. If uV and uV then u + vV.
2. u + v = v + uu, vV (commutativity).
3. (u + v) + w = u + (v + w) ∀ u, v, wV (associativity).
4. 0V such that 0 + v = vvV (the zero vector).
5. vV, ∃ uV such that v + u = 0.
6. a⋅(bv) = (ab)⋅vvV and a, bR.
7. 1⋅v = v where 1 ∈ R.
8. a⋅(u + v) = au + avu, vV and aR.
9. (a + b)⋅v = av + bvvV and a, bR.

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 ith row of M by Mi, 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...