# Cofiniteness - Other Examples - Direct Sum

Direct Sum

The elements of the direct sum of modules are sequences where cofinitely many .

The analog (without requiring that cofinitely many are zero) is the direct product.

### Other articles related to "direct sum, direct":

Projection (linear Algebra) - Projections On Normed Vector Spaces
... A given direct sum decomposition of X into complementary subspaces still specifies a projection, and vice versa ... If X is the direct sum X = U ⊕ V, then the operator defined by P(u + v) = u is still a projection with range U and kernel V ... The relation I = P + (I − P) implies X is the direct sum Ran(P) ⊕ Ran(I − P) ...
Semisimple
... into 'simple' ('irreducible') parts, that fit together in the cleanest way (by direct sum) ... Briefly, "semisimple = direct sum of simple", or equivalently "completely reducible = direct sum of irreducible" ... module is one in which each submodule is a direct summand ...
Maschke's Theorem - Reformulation and The Meaning
... representation built from irreducible subrepresentations using the direct sum operation? In the module-theoretic language, is an arbitrary module semisimple? In ... since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents) ... theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities ...
Internal Direct Sum
... An internal direct sum is simply a direct sum of subobjects of an object ... For example, the real vector space R2 = {(x, y) x, y ∈ R} is the direct sum of the x-axis {(x, 0) x ∈ R} and the y-axis {(0, y) y ∈ R}, and ... More generally, given a vector space V and two subspaces U and W, V is the (internal) direct sum U ⊕ W if U + W = {u + w u ∈ U, w ∈ W} = V, and if u + w = 0 with u ∈ U and w ∈ W, then u = w = 0 ...
Split-biquaternion - Direct Sum of Two Quaternion Rings
... The direct sum of the division ring of quaternions with itself is denoted ... The product of two elements and is in this direct sum algebra ... on the basis of the Proposition it is apparent that Clifford biquaternions split into the direct sum of real quaternions ...

### Famous quotes containing the words sum and/or direct:

Genius is no more than childhood recaptured at will, childhood equipped now with man’s physical means to express itself, and with the analytical mind that enables it to bring order into the sum of experience, involuntarily amassed.
Charles Baudelaire (1821–1867)

The frequency of personal questions grows in direct proportion to your increasing girth. . . . No one would ask a man such a personally invasive question as “Is your wife having natural childbirth or is she planning to be knocked out?” But someone might ask that of you. No matter how much you wish for privacy, your pregnancy is a public event to which everyone feels invited.
Jean Marzollo (20th century)