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  2. Normed vector space - Wikipedia

    en.wikipedia.org/wiki/Normed_vector_space

    Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm , but it is not complete for this norm.

  3. Matrix norm - Wikipedia

    en.wikipedia.org/wiki/Matrix_norm

    Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.

  4. Norm (mathematics) - Wikipedia

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

    Every (real or complex) vector space admits a norm: If = is a Hamel basis for a vector space then the real-valued map that sends = (where all but finitely many of the scalars are ) to | | is a norm on . [9] There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

  5. Polarization identity - Wikipedia

    en.wikipedia.org/wiki/Polarization_identity

    In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm.

  6. Kolmogorov's normability criterion - Wikipedia

    en.wikipedia.org/wiki/Kolmogorov's_normability...

    Kolmogorov's normability criterion — A topological vector space is normable if and only if it is a T 1 space and admits a bounded convex neighbourhood of the origin. Because translation (that is, vector addition) by a constant preserves the convexity, boundedness, and openness of sets , the words "of the origin" can be replaced with "of some ...

  7. Examples of vector spaces - Wikipedia

    en.wikipedia.org/wiki/Examples_of_vector_spaces

    Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.

  8. Space (mathematics) - Wikipedia

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

    More generally, a vector space over a field also has the structure of a vector space over a subfield of that field. Linear operations, given in a linear space by definition, lead to such notions as straight lines (and planes, and other linear subspaces); parallel lines; ellipses (and ellipsoids).

  9. Strictly convex space - Wikipedia

    en.wikipedia.org/wiki/Strictly_convex_space

    In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at ...