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### III.62 Normed Spaces and Banach Spaces

It is often useful to approximate a function f by a polynomial P. For example, if you are designing a pocket calculator and want it to calculate LOGARITHMS [III.25 §4], you cannot expect it to do so exactly, since a calculator cannot handle infinitely many digits, so instead you will get it to calculate a different function P(x) that approximates log(x) well. Polynomials are a good choice, because they can be built up from the basic operations of addition and multiplication. This idea raises two questions: which functions can you hope to approximate, and what counts as a good approximation?

Clearly, the answer to the second question determines the answer to the first, but there is no single right answer ...

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