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A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.

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What are the major differences between real and complex Banach space?

In optimization, you typically have a function $f \colon X \to \mathbb{R}$ which you are going to minimize. In order to apply first-order optimality conditions or first-order methods, you would like t …
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