**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.

Read more about this topic: Cofiniteness, Other Examples

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

... 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) ...

... 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 ...

... 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 ...

**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 ...

**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)