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I am reading a book on differential calculus in normed linear space and something escapes me about how to define differentiability.

I consider $f : U\to E$ where U is an open subset of $V$ and $V$ and $E$ are two Banach spaces.

We say that $f$ is differentiable on $U$ if there exists $\Lambda\in L(V,E)$ such that $f(x+h) - f(x) - \Lambda(h) = o(h)$ . We want to show that this implies continuity

Consider $\epsilon>0$, take $0<\delta<1$ such that if $\lVert h\rVert_{V}\leq\delta$ then $x+h\in U$ and $\lVert f(x+h) - f(x) - \Lambda(h)\rVert_{E}\leq \lVert h\rVert_{V}\quad (1) $

This requirement seems clear for me. In fact it relies on the notion of tangent in Banach space and it is explained in the text as follows : $f$ and $g$ are said to be tangent to one another at $x_0$ if $\forall\epsilon>0, \exists r>0 : \lVert h\rVert < r \implies \lVert f(x_0+h) - g(x_0+h)\rVert\leq\epsilon\lVert h\rVert$

And in the definition of the differentiability introduce in the text this "tangent property" is satisfied so this inequality makes sense.

Now, here is the point I don't understand after $(1)$ : Consequently, $\lVert f(x+h) - f(x)\rVert_{E}\leq (\lVert\Lambda\rVert_{E} +1)\lVert h\rVert_{V}$

And I don't see where does it come from, someone has an idea ? Thank you a lot !

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1 Answer 1

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Using $\lVert f(x+h) - f(x) - \Lambda(h)\rVert_{E}\leq \lVert h\rVert_{V}\quad (1) $ and triangle inequality we get $\lVert f(x+h) - f(x) \leq \lVert h\rVert_{V} +\lVert\Lambda(h)\rVert_{E}$ and $\|\Lambda (h)\|_E\leq \|\Lambda \|_E \|h\|_V$.

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  • $\begingroup$ Thank you a lot for your answer. We have $0\leq\lVert f(x+h) - f(x) - \Lambda(h)\rVert_{E}\leq\lVert h\rVert_{V}$ So how can we insure that by the triangle inequality we don't have this issue $\lVert f(x+h) - f(x)\rVert_{E} + \lVert\Lambda(h)\rVert_{E}\geq \lVert h\rVert_{V}$ ? And also I don't see how you get $\lVert\Lambda(h)\rVert_{E}\leq\lVert\Lambda\rVert_{E}\lVert h\rVert_{V}$ ? Thank you $\endgroup$
    – coboy
    Dec 22, 2022 at 23:36
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    $\begingroup$ For the first question write $f(x+h) - f(x) $ as $[f(x+h) - f(x) - \Lambda(h)]+\Lambda(h)$ and apply triangle inequlaity. For the second question $\|Tx\|\leq ||T\|\|x\|$ for any bounded operator on a normed linear space. This is an easy consequence of the definition of norm of an operator. @coboy $\endgroup$ Dec 22, 2022 at 23:47
  • $\begingroup$ Thank you a lot it is clear now ! $\endgroup$
    – coboy
    Dec 23, 2022 at 0:05

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