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Differentiation can also be defined to functions of several variables (for example, or even , where is an infinite-dimensional vector space). If X {\displaystyle X} is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if f = ( f 1 , f 2 , f 3 , …
In mathematics, a Schauder basis or countable basis is similar to the usual basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums.
For example, the complex numbers C form a two-dimensional vector space over the real numbers R. Likewise, the real numbers R form a vector space over the rational numbers Q which has (uncountably) infinite dimension, if a Hamel basis exists. [b] If V is a vector space over F it may also be regarded as vector space over K. The dimensions are ...
More generally, if W is a linear subspace of a (possibly infinite dimensional) vector space V then the codimension of W in V is the dimension (possibly infinite) of the quotient space V/W, which is more abstractly known as the cokernel of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition
In the infinite-dimensional case, the existence of infinite bases, often called Hamel bases, depends on the axiom of choice. It follows that, in general, no base can be explicitly described. [16] For example, the real numbers form an infinite-dimensional vector space over the rational numbers, for which no specific basis is known.
bicomplex numbers: a 4-dimensional vector space over the reals, 2-dimensional over the complex numbers, isomorphic to tessarines. multicomplex numbers: 2 n-dimensional vector spaces over the reals, 2 n−1-dimensional over the complex numbers; composition algebra: algebra with a quadratic form that composes with the product
In general, given two complex topological vector spaces X and Y and an open set U ⊂ X, there are various ways of defining holomorphy of a function f : U → Y.Unlike the finite dimensional setting, when X and Y are infinite dimensional, the properties of holomorphic functions may depend on which definition is chosen.
The vector space is said to be the algebraic direct sum (or direct sum in the category of vector spaces) when any of the following equivalent conditions are satisfied: The addition map S : M × N → X {\\displaystyle S:M\\times N\\to X} is a vector space isomorphism .