# Weak closure points not attainable as limits of sequences

Where in this post we discover an uncanny property of the weak topology: the points of the weak closure of a set cannot always be attained as limits of elements of the set. Naturally, the w-closure of a set is weakly closed. The so-called weak sequential closure of a set, on the other hand, is the set of cluster points of sequences made with elements from that set. The new set is not, however, weakly sequentially closed, which means that there may arise new cluster points; we may, in fact, have to take the weak sequential closure trans-finitely many time to obtain a set which is weakly sequentially closed – still, this may not be weakly closed. Continue reading →

# Linearisation bounds for smooth mutlivalued functions

How far is a function $f$ from its linearisation? Typically, one would assume that $f$ is twice continuously differentiable and use the following second-order version of the mean value theorem:

$f(y) = f(x) + \langle J f(x),y-x\rangle + \int_0^1 (1-\tau) (y-x)'\nabla^2 f(x+\tau (y-x))(y-x)\mathrm{d}\tau$

This is typically used in the context of linearisation of nonlinear dynamical systems as in [Sec. 2.5.1.3, 1]. The requirement that $f$ is twice continuously differentiable, can, however, be reduced to $f$ being continuously differentiable with Lipschitz gradient. Continue reading →

Updating the Cholesky factorization of $A'A=LL'$ when one or more columns are added to or removed from matrix $A$ can be done very efficiently obviating the re-factorization from scratch. Continue reading →

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