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I have a question about the differential of a function between two normed spaces. It is a simple question about the definition.

In my textbook from my university, the definition is as follows:

Let $E$ an $F$ be normed spaces, $U \subset E$ open and $f:U \rightarrow F$. f is called differentiable in $a \in U$, if there exists a continuous, linear map $L:E \rightarrow F$, for which, with $||h||$ sufficient smal,

$\qquad$ $\qquad$ $\qquad$$\qquad$$\qquad$$f(a+h) = f(a) + L*h +o(h), \qquad h \rightarrow 0$

Could it be this is wrong?

If I search for it online, I often find $L(h)$ instead of $L*h$

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It's probably a typo -- or a "brain-o". For many (e.g., finite dimensional) spaces, linear maps are the same as "multiply by a matrix", once you've picked a basis, so people often identify a linear map with its matrix representation, at which point $L(h)$ and $L * h$ become almost synonyms, at least if you treat "*" as denoting multiplication.

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  • $\begingroup$ People use product notation even without introducing coordinates. It's really much more convenient than rigidly treating objects of (multi)linear algebra as specific kinds of functions. $\endgroup$ – Hurkyl Jun 6 '15 at 15:52
  • $\begingroup$ Agreed (and understood). It's also (as shown by this question) potentially confusing. :) $\endgroup$ – John Hughes Jun 6 '15 at 16:20

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