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Linear Algebra
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
A nonempty subset W⊂V $ W \subset V $ is a linear subspace of V if and only if it is closed with respect to both operations: vector addition and multiplication by a scalar, i.e., u,v∈W⇒u+v∈Wα∈R(C),u∈W⇒αu∈W $$ \begin{aligned} \boldsymbol{u},\boldsymbol{v}\in \ W \quad \Rightarrow \quad \boldsymbol{u}+ \boldsymbol{v}\in \ W \\ \alpha \in \ \mathbb R ( \mathbb C ), \boldsymbol{u}\in \ W \quad \Rightarrow \quad \alpha \boldsymbol{u}\in \ W \end{aligned} $$
Some Basic Concepts in Functional Analysis
Published in P.N. Natarajan, Functional Analysis and Summability, 2020
Let S be a non-empty subset of a linear space X. If S itself is a linear space with respect to the operations of addition and scalar multiplication defined in X, then S is called a “linear subspace” or simply a “subspace” of X.
Tensor methods for finding approximate stationary points of convex functions
Published in Optimization Methods and Software, 2022
G. N. Grapiglia, Yurii Nesterov
For simplicity, assume that and . Given an approximation for the solution of (4), we consider p-order methods that compute trial points of the form , where the search direction is the solution of an auxiliary problem of the form with , and q>1. Denote by the set of all stationary points of function and define the linear subspace More specifically, we consider the class of p-order tensor methods characterized by the following assumption.
On unconstrained optimization problems solved using the canonical duality and triality theories
Published in Optimization, 2020
The above text shows that, at least in [8, Sect. 3.5], is a function space like . Of course, F being a linear function on , has to be a linear subspace endowed we the trace topology. A linear functional f defined on a topological vector space U is Gâteaux differentiable if and only if f is continuous, in which case for every moreover, it is not possible to speak about ‘the Legendre conjugate of F’. So, (A2) has not a mathematical meaning. Moreover, in order to speak about and in (A3), one needs and be at least algebraically open (convex) subsets of and respectively. It is clear that the concerned spaces are not one-dimensional.
New versions of Newton method: step-size choice, convergence domain and under-determined equations
Published in Optimization Methods and Software, 2020
Key component in Newton method is the auxiliary convex optimization sub-problem, involving the linear constraint. Note that the constraint describes either a linear subspace, or the empty set. The classical result below (which goes back to Banach, see [14,18,20]) guarantees solvability of the linear Equation (7) and gives an estimate of its solution. We prefer to provide the direct proof of the result because it is highly clear and short in finite-dimensional case. Suppose that spaces are equipped with some norms, the dual norms are denoted (for a linear functional c, associated with the vector of the same dimension, ). Operator norm is subordinate with the vector norms, e.g. for we have . In most cases we do not specify vector norms; dual norms are obvious from the context. The adjoint operator is identified with matrix .