# derivative

## Primary tabs

Defines:
directional derivative,Fr\'echet derivative
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### Norm on L(V,W)

I mention in the article above that L(V,W)
can be considered a Banach space itself.

The fact that it is a vector space is trivial.
However, I'm not sure what the canonical norm
for that space is. Perhaps the L^p norm?
What are some other commonly used norms that
turn L(V,W) into a normed space? Do they
generate the same topology as the L^p norm?

### Re: Norm on L(V,W)

After discussing this with a friend, it
looks like the natural norm for this space
is the "operator norm" or "induced norm".

### Possible error in formula

Under Linearity, it seems like there is a typo in the formula, bg(x) becomes bf'(x) which doesn't seem right but its been a long time since I studied this sort of thing so I could be wrong.

Dwayne

### Re: Possible error in formula

You're right, thanks for pointing this out. BTW, this type of note is better filed as a correction.

### Derivative as Linear Approximation

I understand that the Frechet derivative is a linear approximation in the sense that it is a linear transformation which approximates a function between Banach spaces.

But, I have read in multiple texts that it is the "best" linear approximation. In what sense in the Frechet derivative the "best" linear approximation as opposed to just some linear approximation?