Books:Analysis - Part 1: Elements - Krzysztof Maurin
From Maths
Alec's progress
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Key
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Mistakes
- p59 - the line in the middle that reads: "[ilmath]x-x_0:=h[/ilmath], [ilmath]f(x+h)-f(x)=f'(x)h+r(x,h)[/ilmath]" should read:
- "[ilmath]x-x_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=x-x_0[/ilmath] to [ilmath]h=x_0-x[/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)
- p154 - The book suggests that "continuity at a particular point" is equivalent to a linear map between two normed spaces mapping null sequences to bounded sequences, while not wrong (as such a map is also continuous at every point) it only proves that it is continuous at zero (not some arbitrary given point), another book[1] suggests that continuity at 0 is what was intended. See Equivalent conditions for a linear map between two normed spaces to be continuous everywhere
