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Advanced Mathematics for Applications by Andrea Prosperetti

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21

Linear Operators in Infinite-Dimensional Spaces

 

Many of the examples in Part I of this book are built around the idea of solving a linear equation by expanding the solution on a basis of eigenfunctions of a suitable operator. At a superficial glance, even integral transform techniques appear to be based on the same general idea although function such as eikx, for example, cannot be regarded as a true eigenfunction of a differential operator.

Anyone familiar with the analogous method for dealing with a linear algebraic finite-dimensional system, Au = f, cannot but be struck by the similarity of the approaches.

There is an important difference, however, and that resides in the infinite dimensionality of the space in which the solution ...

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