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  2. Complex number - Wikipedia

    en.wikipedia.org/wiki/Complex_number

    More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .

  3. Linear complex structure - Wikipedia

    en.wikipedia.org/wiki/Linear_complex_structure

    The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...

  4. Resolvent set - Wikipedia

    en.wikipedia.org/wiki/Resolvent_set

    In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism .

  5. Outline of algebra - Wikipedia

    en.wikipedia.org/wiki/Outline_of_algebra

    Fundamental theorem of algebra – states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equations – equality of two mathematical expressions

  6. List of lemmas - Wikipedia

    en.wikipedia.org/wiki/List_of_lemmas

    1.3 Linear algebra. 1.4 Group theory. ... (several complex variables) ... An example of a covering described by the Knaster–Kuratowski–Mazurkiewicz lemma.

  7. Linear algebra - Wikipedia

    en.wikipedia.org/wiki/Linear_algebra

    In multilinear algebra, one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the dual space , the vector space V* consisting of linear maps f : V → F where F is the field of scalars.

  8. Fredholm alternative - Wikipedia

    en.wikipedia.org/wiki/Fredholm_alternative

    The Fredholm alternative is the statement that, for every non-zero fixed complex number, either the first equation has a non-trivial solution, or the second equation has a solution for all (). A sufficient condition for this statement to be true is for K ( x , y ) {\displaystyle K(x,y)} to be square integrable on the rectangle [ a , b ] × [ a ...

  9. Space (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Space_(mathematics)

    Every complex linear space is also a real linear space (the latter underlies the former), since each complex number can be specified by two real numbers. For example, the complex plane treated as a one-dimensional complex linear space may be downgraded to a two-dimensional real linear space. In contrast, the real line can be treated as a one ...

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