Books:Analysis  Part 1: Elements  Krzysztof Maurin
From Maths
Alec's progress
Book progress  

Key

Mistakes
 p59  the line in the middle that reads: "[ilmath]xx_0:=h[/ilmath], [ilmath]f(x+h)f(x)=f'(x)h+r(x,h)[/ilmath]" should read:
 "[ilmath]xx_0:=h[/ilmath], [ilmath]f(x_0+h)f(x)=f'(x_0)h+r(x_0,h)[/ilmath]" otherwise we're dealing with [ilmath]f(2x+x_0)f(x)[/ilmath] on the LHS and there are problems. If you swap [ilmath]h=xx_0[/ilmath] to [ilmath]h=x_0x[/ilmath] the [ilmath]f(x_0)f(x)[/ilmath] part of the definition is negative what it ought to be; although the linear map's (the differential) argument is negated so it still sort of works out, either way replacing [ilmath]x[/ilmath] with [ilmath]x_0[/ilmath] is the easiest and most straightforward solution. This is inline with the definition of differentiability (Banach Space) given on p161
 p149  wrongly (and explicitly) states that a norm may be a function of the form [ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{C} [/ilmath], see the norm page (there's a warning note on the first line of the definition section)