Math - Analysis - Norms of Functions

**cosmonavt** · July 24th 2012, 01:23 AM

Hello guys,

I know that norm of a vector is its magnitude. But what is the norm of a function? How do we visualize it? Been searching on the web but can't seem to understand.

Thanks

**Plato** · July 24th 2012, 04:22 AM

Originally Posted by cosmonavt

I know that norm of a vector is its magnitude. But what is the norm of a function? How do we visualize it? Been searching on the web but can't seem to understand.

You must redirect your concept of norm. You say "I know that norm of a vector is its magnitude". Well the truth is that length is a norm but a norm is not necessarily length.

A norm is a function from the vector space to the underlying field: $\|:X\to R$ .
It must satisfy three rules:
1) $\|x\|\ge 0$ and is zero if and only if $x=0$ ,
2) if $a$ is a scalar then $\|ax\|=|a|\|x\|$ ,
3) $\|x+y\|\le \|x\|+\|y\|$ .

**cosmonavt** · July 24th 2012, 06:08 AM

What is this underlying field? The set of Real Numbers right?

Also, like there is a vector space $R^{d}$ , is $L^{p}$ a function space?

Like there is Cautchy-Shwarz Inequality for vectors, do we have the equivalent Holder's Inequality for functions?

If this is the Holder's Inequality: $\|fg\|_1 \le \|f\|_p \|g\|_q$

don't $\|f\|_p$ and $\|g\|_p$ mean the norm of function f and norm of function g respectively?

That's my question. I know that the norm of a vector in a vector space is its length (usually), but what exactly is the norm of a function in a function space?

If I am going wrong, correct me.

**HallsofIvy** · July 24th 2012, 06:48 AM

The norm is part of the definition of $L^p$ . " $L^p$ " is the set of functions, f, such that $|f|^2$ is integrable and the norm is defined as $\sqrt[p]{\int |f(x)|^p dx}$ .

**cosmonavt** · July 24th 2012, 07:14 AM

Originally Posted by HallsofIvy

The norm is part of the definition of $L^p$ . " $L^p$ " is the set of functions, f, such that $|f|^2$ is integrable and the norm is defined as $\sqrt[p]{\int |f(x)|^p dx}$ .

That's new knowledge for me. Thanks. However, when we say the square of the modulus of $f$ --- $|f|^2$ ---- does it yield a numerical value? Isn't $f$ itself a function in terms of $x$ (or some other variable). That way $|f|^2$ should yield an expression in terms of $x$ . Correct?

**HallsofIvy** · July 24th 2012, 07:27 AM

No. There are two errors in what I wrote before: first, I should have said "such that $|f|^p$ is integrable", not $|f|^2$ and, second, I should have specified the interval of integration. If you are dealing with the simplest case, functions such that $|f|^2$ is integrable over all real numbers, then the norm is defined as
$\sqrt[p]{\int_{-\infty}^\infty |f(x)|^pdx}$ .

Of course, we can talk, more generally, about $L^p(C)$ where C is some set (typically a closed and bounded interval of the real numbers or some compact subset of the complex numbers) and then the integral is over the et C.

I probably accidently wrote " $|f|^2$ " because the most important case is p= 2, the "square integrable functions". In that case, if f and g are both "square integrable" over set C, it can be shown that the product fg is C and we can define the inner product $\int_C f(x)g(x)dx$ and have, not just a "normed space" but an "inner product space".

**cosmonavt** · July 24th 2012, 08:42 AM

Got that. But still excuse my inadequacy. How do we visualize a norm of a function? Suppose we have a function $f(x)=x^2+2x+1$ . What does the norm show? Like a norm of a vector shows its length, what does the norm of this function show? Also, do we ever need to calculate these norms by integration in analysis?

I am actually a high school student (not a university student) and been reading up on vector spaces only recently.

**emakarov** · July 24th 2012, 09:54 AM

Originally Posted by cosmonavt

How do we visualize a norm of a function?

Ha. Visualizing in mathematics is overrated. Consider the following joke.

In modern mathematics, algebra has become so important that numbers will soon only have symbolic meaning.

If this is true about numbers, then how much more about vectors... Here is another joke about visualization.

A Mathematician and an Engineer attend a lecture by a Physicist. The topic concerns Kulza-Klein theories involving physical processes that occur in spaces with dimensions of 9, 12 and even higher. The Mathematician is sitting, clearly enjoying the lecture, while the Engineer is frowning and looking generally confused and puzzled. By the end the Engineer has a terrible headache. At the end, the Mathematician comments about the wonderful lecture. The Engineer says "How do you understand this stuff?"
Mathematician: "I just visualize the process"
Engineer: "How can you POSSIBLY visualize something that occurs in 9-dimensional space?"
Mathematician: "Easy, first visualize it in N-dimensional space, then let N go to 9"

Generally, you can make any set: numbers, matrices, functions, plane translations, buses, people and so on — into a vector space if you give a proper definition of operations. Similarly, you can define a norm on objects like these. Ultimately, it can be nothing more than a function satisfying axioms in post #2. There may not be an obvious way to visualize it.

One way to define a norm of $f:\mathbb{R}\to\mathbb{R}$ is $\|f\|=\sup_{x\in\mathbb{R}}|f(x)|$ . Here, the geometric sense is clear: e.g., functions whose norm does not exceed 1 never deviate from 0 by more than 1. In the space $L^1$ , $\|f\|=\int_{-\infty}^{\infty}|f(x)|\,dx$ . Here the norm is the area between the x-axis and the graph of $f(x)$ . Note, by the way, that vectors in n-dimensional vector spaces over, say, $\mathbb{R}$ , can also be considered as functions from {1, ..., n} to $\mathbb{R}$ . Besides the Euclidean norm, an important norm is $\|x\|_1$ . If x is such n-dimensional vector, then $\|x\|_1=|x(1)|+\dots+|x(n)|$ . This norm is called Manhattan distance.

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