By F. M. Hall

ISBN-10: 0521070554

ISBN-13: 9780521070553

ISBN-10: 0521298628

ISBN-13: 9780521298629

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**Example text**

In many situations continuous functions and the Riemann integral will be too restrictive and we will have to extend the notion of the Lebesgue integral to vector-valued functions. The most commonly used generalization of the Lebesgue integral is oﬀered by the Bochner integral which is now brieﬂy discussed (see [172, pp. 130–136], [100, pp. 58–92], or [12, pp. 5–15]). The starting point in the deﬁnition of the Lebesgue integral is the notion of measurability of a function. The standard deﬁnition used in the real function theory cannot be used here and is replaced by the following construction carried out, for the sake of generality, in Rn .

Any surjective A ∈ L(X, Y ) is an open mapping. One of the most often used consequences of this theorem is the Bounded Inverse Theorem. 17. If A ∈ L(X, Y ) is such that KerA = {0} and ImA = Y , then A−1 ∈ L(Y, X). The corollary follows as the assumptions on the kernel and the image ensure the existence of a linear operator A−1 deﬁned on the whole Y . The operator A−1 is continuous by the Open Mapping Theorem, as the preimage of any open set in X through A−1 , that is, the image of this set through A, is open.

In other words ∂t φ1 (t, ·) = u(n) (t). 69), t u(n−1) (z)−u(n−1) (α) dz = u(n−2) (t)−u(n−2) (α)−(t−α)u(n−1) (α). φ2 (t, ·) = α Again, φ2 (t, s) is measurable on I × Ω, but now, because φ1 (t, s) is absolutely continuous in t for each s, we obtain that ∂t φ2 (t, s) = φ1 (t, s) at all points of I × Ω. Proceeding in this way, we ﬁnally obtain t φn (t, s) = φn−1 (z, s)dz, α where φn (t, ·) = u(t) − n−1 (t k=0 − α)k (k) u (α). k! Because α is ﬁxed, we can take any representation u(k) (α) and obtain in this way the desired representation of u, namely, φ(t, s) = φn (t, s) + n−1 (t k=0 − α)k (k) [u (α)](s).

### An introduction to abstract algebra, by F. M. Hall

by Richard

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