Coverage for local/lib/python2.7/site-packages/sage/categories/vector_spaces.py : 33%
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r""" Vector Spaces """ #***************************************************************************** # Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu> # William Stein <wstein@math.ucsd.edu> # 2008 Teresa Gomez-Diaz (CNRS) <Teresa.Gomez-Diaz@univ-mlv.fr> # 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #****************************************************************************** from sage.categories.category import Category from sage.categories.category_types import Category_module from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring from sage.categories.cartesian_product import CartesianProductsCategory from sage.categories.dual import DualObjectsCategory from sage.categories.tensor import TensorProductsCategory from sage.categories.fields import Fields from sage.categories.modules import Modules from sage.categories.modules_with_basis import ModulesWithBasis _Fields = Fields()
class VectorSpaces(Category_module): """ The category of (abstract) vector spaces over a given field
??? with an embedding in an ambient vector space ???
EXAMPLES::
sage: VectorSpaces(QQ) Category of vector spaces over Rational Field sage: VectorSpaces(QQ).super_categories() [Category of modules over Rational Field] """ @staticmethod def __classcall_private__(cls, K, check=True): """ INPUT:
- `K` -- a field - ``check`` -- a boolean (default: True) whether to check that `K` is a field.
EXAMPLES::
sage: VectorSpaces(QQ) is VectorSpaces(QQ, check=False) True
By default, it is checked that ``K`` is a field::
sage: VectorSpaces(ZZ) Traceback (most recent call last): ... ValueError: base must be a field or a subcategory of Fields(); got Integer Ring
With ``check=False``, the check is disabled, possibly enabling incorrect inputs::
sage: VectorSpaces(ZZ, check=False) Category of vector spaces over Integer Ring """ (isinstance(K, Category) and K.is_subcategory(_Fields))): " got {}".format(K))
def __init__(self, K): """ EXAMPLES::
sage: VectorSpaces(QQ) Category of vector spaces over Rational Field sage: VectorSpaces(ZZ) Traceback (most recent call last): ... ValueError: base must be a field or a subcategory of Fields(); got Integer Ring
TESTS::
sage: C = QQ^10 # vector space sage: TestSuite(C).run() sage: TestSuite(VectorSpaces(QQ)).run() """
def __call__(self, x): """ Try to coerce ``x`` into an object of this category
EXAMPLES::
sage: VectorSpaces(QQ)(ZZ^3) Vector space of dimension 3 over Rational Field
""" V = V.change_ring(self.base_field()) except (TypeError, AttributeError) as msg: raise TypeError("%s\nunable to coerce x (=%s) into %s"%(msg,x,self))
def base_field(self): """ Returns the base field over which the vector spaces of this category are all defined.
EXAMPLES::
sage: VectorSpaces(QQ).base_field() Rational Field """
def super_categories(self): """ EXAMPLES::
sage: VectorSpaces(QQ).super_categories() [Category of modules over Rational Field] """
def additional_structure(self): r""" Return ``None``.
Indeed, the category of vector spaces defines no additional structure: a bimodule morphism between two vector spaces is a vector space morphism.
.. SEEALSO:: :meth:`Category.additional_structure`
.. TODO:: Should this category be a :class:`CategoryWithAxiom`?
EXAMPLES::
sage: VectorSpaces(QQ).additional_structure() """
class ParentMethods: pass
class ElementMethods: pass
class WithBasis(CategoryWithAxiom_over_base_ring):
_call_ = ModulesWithBasis.__dict__["_call_"]
def is_abelian(self): """ Return whether this category is abelian.
This is always ``True`` since the base ring is a field.
EXAMPLES::
sage: VectorSpaces(QQ).WithBasis().is_abelian() True """
class CartesianProducts(CartesianProductsCategory): def extra_super_categories(self): r""" The category of vector spaces with basis is closed under Cartesian products::
sage: C = VectorSpaces(QQ).WithBasis() sage: C.CartesianProducts() Category of Cartesian products of vector spaces with basis over Rational Field sage: C in C.CartesianProducts().super_categories() True """
class TensorProducts(TensorProductsCategory): def extra_super_categories(self): r""" The category of vector spaces with basis is closed under tensor products::
sage: C = VectorSpaces(QQ).WithBasis() sage: C.TensorProducts() Category of tensor products of vector spaces with basis over Rational Field sage: C in C.TensorProducts().super_categories() True """
class DualObjects(DualObjectsCategory):
def extra_super_categories(self): r""" Returns the dual category
EXAMPLES:
The category of algebras over the Rational Field is dual to the category of coalgebras over the same field::
sage: C = VectorSpaces(QQ) sage: C.dual() Category of duals of vector spaces over Rational Field sage: C.dual().super_categories() # indirect doctest [Category of vector spaces over Rational Field] """
class CartesianProducts(CartesianProductsCategory): def extra_super_categories(self): r""" The category of vector spaces is closed under Cartesian products::
sage: C = VectorSpaces(QQ) sage: C.CartesianProducts() Category of Cartesian products of vector spaces over Rational Field sage: C in C.CartesianProducts().super_categories() True """
class TensorProducts(TensorProductsCategory): def extra_super_categories(self): r""" The category of vector spaces is closed under tensor products::
sage: C = VectorSpaces(QQ) sage: C.TensorProducts() Category of tensor products of vector spaces over Rational Field sage: C in C.TensorProducts().super_categories() True """ |