Search results
Results from the WOW.Com Content Network
If is a function of real numbers with values in a Hilbert space , then the derivative of at a point can be defined as in the finite-dimensional case: ′ = (+) (). Most results of the finite-dimensional case also hold in the infinite-dimensional case too, with some modifications.
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 ...
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 .
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.
is constant-recursive because it satisfies the linear recurrence = +: each number in the sequence is the sum of the previous two. [2] Other examples include the power of two sequence 1 , 2 , 4 , 8 , 16 , … {\displaystyle 1,2,4,8,16,\ldots } , where each number is the sum of twice the previous number, and the square number sequence 0 , 1 , 4 ...
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers.Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers.
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, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6. Some properties of this problem are: If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.