# IUTAM Symposium on Advanced Optical Methods and Applications by Alexis Lagarde

By Alexis Lagarde

**Read or Download IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics: proceedings of the IUTAM Symposium held in Futuroscope, Poitiers, France, August 31st - September 4th PDF**

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1) is said to be bounded. 1 A linear operator is continuous iff it is bounded. 2 Let X,Y be normed linear spaces and A be a linear operator from X into Y. If X is finite dimensional, then A is continuous. Proof. Let dim X = N and suppose set of elements in X. If we can write is a linearly independent uniquely. 1 to give where Thus so that A is continuous. 4 Let X be a finite dimensional normed linear space and a linear functional on X. Show that is continuous. There are classes of differential and integral operators that are continuous, as given by the following problems.

1 Suppose is a bounded domain, and If then and satisfy Proof. 7) . 9) shows that if is a Cauchy sequence in then it will also be a Cauchy sequence in and 2. Introduction to Metric Spaces 56 Moreover, if is an equivalence class of Cauchy sequences in then this will also be an equivalence class in Thus we can proceed to the limit in the inequality and say that if then and We can show this schematically as in Fig. 2. 7) states that it is continuous (or bounded). We can make this formal. 4 We say that the normed space X is imbedded in the normed space Y, and write this if X is a subspace of Y, and the operator I from X to Y defined by for all is continuous.

7) . 9) shows that if is a Cauchy sequence in then it will also be a Cauchy sequence in and 2. Introduction to Metric Spaces 56 Moreover, if is an equivalence class of Cauchy sequences in then this will also be an equivalence class in Thus we can proceed to the limit in the inequality and say that if then and We can show this schematically as in Fig. 2. 7) states that it is continuous (or bounded). We can make this formal. 4 We say that the normed space X is imbedded in the normed space Y, and write this if X is a subspace of Y, and the operator I from X to Y defined by for all is continuous.