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A set is relatively open iff it is equal to its relative interior. Note that when aff ( S ) {\displaystyle \operatorname {aff} (S)} is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed.
The interior, boundary, and exterior of a set together partition the whole space into three blocks (or fewer when one or more of these is empty): = , where denotes the boundary of . [3] The interior and exterior are always open, while the boundary is closed.
It states that a decreasing nested sequence () of non-empty, closed and bounded subsets of has a non-empty intersection. This version follows from the general topological statement in light of the Heine–Borel theorem , which states that sets of real numbers are compact if and only if they are closed and bounded.
Every Jordan block J i corresponds to an invariant subspace X i. Symbolically, we put = = where each X i is the span of the corresponding Jordan chain, and k is the number of Jordan chains. One can also obtain a slightly different decomposition via the Jordan form.
As a result, the empty set is the unique initial object of the category of sets and functions. The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps.
An idempotent linear operator is a projection operator on the range space along its null space . P {\displaystyle P} is an orthogonal projection operator if and only if it is idempotent and symmetric .
A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets [2] (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: [ 3 ]
Let () (that is, a n × n complex matrix) and () be the change of basis matrix to the Jordan normal form of A; that is, A = C −1 JC.Now let f (z) be a holomorphic function on an open set such that ; that is, the spectrum of the matrix is contained inside the domain of holomorphy of f.