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derivative

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

Mathematics Subject Classification

26B05 no label found46G05 no label found26A24 no label found

Comments

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?

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

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

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

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?

PlanetMath article: Non-Newtonian calculus.

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