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Lecture 24 - Mar 16

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Normed Linear Space

Let be a vector space over or . A norm on is a map

such that for all and :

  1. .
  2. .
  3. .

A vector space equipped with a norm is called a normed linear space.

Examples

  1. or , with
  1. , with
  1. , with

Metric From Norm

If is a normed linear space, then

defines a metric on .

Normed linear spaces that are complete are called Banach spaces.

If a norm comes from an inner product, and the space is complete, it is called a Hilbert space.

Spaces

Let be a measure space. For , define

Define

Then is a vector space (check), and is the natural norm.

Holder’s Inequality

Let and let be the conjugate exponent:

If and , then

In particular, for fixed ,

defines a linear map.

Proof

If or , the result is trivial.

First assume . By Young’s inequality,

Integrating,

For general , set

Then , so

Hence

Minkowski’s Inequality

If and , then

Proof

For , this is the triangle inequality under the integral.

Now assume . Let satisfy , so

If a.e., done. Otherwise,

Integrate:

Apply Holder to each term:

So

Now,

Therefore

which is Minkowski’s inequality.

Thus, is a normed linear space (concluded from above inequalities).

These two inequalities are the key tools for proving that is a norm on .

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