By Rodney Coleman

This ebook serves as an advent to calculus on normed vector areas at a better undergraduate or starting graduate point. the necessities comprise simple calculus and linear algebra, in addition to a undeniable mathematical adulthood. all of the very important topology and sensible research themes are brought the place necessary.

In its try to exhibit how calculus on normed vector areas extends the fundamental calculus of services of a number of variables, this publication is among the few textbooks to bridge the space among the on hand ordinary texts and excessive point texts. The inclusion of many non-trivial functions of the idea and fascinating workouts presents motivation for the reader.

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**Example text**

Show that if f is constant on the boundary of X , then f has a critical point in the interior of X . 6 Differentiability of the Norm If E is a normed vector space with norm k k, then k k is itself a mapping from E into R and we may consider its differentiability. We will write kxk0 for the differential of the norm at x (if it exists). We should first notice that the norm is never differentiable at the origin. Suppose that k0k0 exists. 0 khk 0 Summing the two limits we obtain 2 D 0, which is clearly a contradiction.

As R2 is complete, R2 is a commutative Banach algebra. E/ is a normed algebra. E/ is a Banach algebra. Here is another example of a normed algebra. Consider the set RŒX of real polynomials in one variable. RŒX is clearly an algebra and has the polynomial P Á 1 for multiplicative identity. x/j; x2Œ0;1 then we obtain a normed algebra. A/ of bounded continuous real-valued functions defined on a metric space, in particular, on a nonempty subset of a normed vector space. A/ becomes a Banach algebra.

Proof. Let us set E D Rn , F D Rm and fix norms on these spaces. Suppose that f is of class C 1 . 2, we know that f is differentiable on O. R/. x/j D 0: It follows that f 0 is continuous at x. Now suppose that f 0 is defined and continuous on O. 5 we know that the partial derivatives of f are defined on O. x/jM D 0; which implies that the partial derivatives are continuous at x. The preceding theorem suggests the following generalization of the notion of class C 1 . If E and F are normed vector spaces and O is open in E, then f W O !