Measure Theory/L2 Inner Product Space

Inner Products

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We have defined, in Lesson 0, the inner product for  . However, when calling something by the name "inner product" we mean to communicate a few properties. We should ensure that our inner product,

 

deserves the name.

Besides justifying the name, out of a sense of principle, these properties will also be useful common manipulations in the proofs of the theorems that we care about -- the most important, for this section, being completeness.

Definition: inner product

Let V be any vector space over  , and let   be any binary function. In all of the definitions below, let   be arbitrary vectors and   an arbitrary real number.

We define   to be symmetric if

 

We define it to be linear in the first coordinate (or just linear for short) if

 

We define it to be positive definite if

  with equality holding if and only if  .

We define   to be an inner product on V if it is symmetric, linear, and positive definite.

Recall that you have already decided what   is when the vector space in question is  , in the previous lesson, Lesson 1.


Exercise 1. L2 Inner Product Is an Inner Product


Show that the   inner product is symmetric and linear. (This should be trivial.)

Also show that for every  , we have  .

Also show that  .

Now consider proving that if   then  . First show that, in fact, there is a function   for which   even though  .

Next recall that, technically,   is a set of equivalence classes. Recalling also a result from a previous section, prove that if   then as an equality of equivalence classes, it follows that  .

Conclude that the   inner product, is an inner product on  .

Normed Space

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Definition: norm

Let V be a vector space and   a function. Let   be an arbitrary vector and   an arbitrary real number, throughout this definition. We say that   is positive definite if

  with equality holding if and only if  

We call it homogeneous if

 

We call it subadditive if

 

(We call the inequality itself,  , the triangle inequality.)

We call any function a norm if it is positive definite, homogeneous, and subadditive. We call V a normed vector space if it has a norm.


Exercise 2. L2 Is a Normed Metric Space


Show that every inner product space is a normed space, and infer that   is a normed space with norm  .

Metric Space

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Definition: metric space

Let X be a set and  . Let  . We say that d is positive definite if

  with equality if and only if  

We say that d is symmetric if

 

We call the inequality

 

the triangle inequality. (If we wish, we may distinguish between the norm and distance "triangle inequality".)

The function d is called a distance metric if it is positive definite, symmetric, and satisfies the triangle inequality. The space X is called a metric space if it has a distance metric.


Exercise 3. L2 Is a Metric Space


Prove that every normed space is a metric space, and infer that   is a metric space with metric  .