SUMMARY / RELATED TOPICS

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that has the following properties: The zero vector, 0, has zero length. ‖ x ‖ ≥ 0, ‖ x ‖ = 0 if and only if x = 0 Multiplying a vector by a positive number changes its length without changing its direction. Moreover, ‖ α x ‖ = | α | ‖ x ‖ for any scalar α; the triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line. ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ for any vectors x and y. The generalization of these three properties to more abstract vector spaces leads to the notion of norm. A vector space on which a norm is defined is called a normed space or normed vector space.

Normed vector spaces are central to the study of linear algebra and functional analysis. A normed vector space is a pair where V is a vector space and ‖ ⋅ ‖ a norm on V. A seminormed vector space is a pair where V is a vector space and p a seminorm on V. We omit p or ‖ ⋅ ‖ and just write V for a space if it is clear from the context what norm we are using. In a more general sense, a vector norm can be taken to be any real-valued function that satisfies the three properties above. A useful variation of the triangle inequality is ‖ x − y ‖ ≥ | ‖ x ‖ − ‖ y ‖ | for any vectors x and y; this shows that a vector norm is a continuous function. Note that property 2 depends on a choice of norm | α | on the field of scalars; when the scalar field is R, this is taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over Q one could take | α | to be the p-adic norm, which gives rise to a different class of normed vector spaces. If is a normed vector space, the norm ‖·‖ induces a metric and therefore a topology on V.

This metric is defined in the natural way: the distance between two vectors u and v is given by ‖u−v‖. This topology is the weakest topology which makes ‖·‖ continuous and, compatible with the linear structure of V in the following sense: The vector addition +: V × V → V is jointly continuous with respect to this topology; this follows directly from the triangle inequality. The scalar multiplication ·: K × V → V, where K is the underlying scalar field of V, is jointly continuous; this follows from the triangle homogeneity of the norm. For any semi-normed vector space we can define the distance between two vectors u and v as ‖u−v‖; this turns the seminormed space into a pseudometric space and allows the definition of notions such as continuity and convergence. To put it more abstractly every semi-normed vector space is a topological vector space and thus carries a topological structure, induced by the semi-norm. Of special interest are complete normed spaces called Banach spaces; every normed vector space V sits as a dense subspace inside a Banach space.

All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology. And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space V is locally compact if and only if the unit ball B = is compact, the case if and only if V is finite-dimensional; the topology of a seminormed vector space has many nice properties. Given a neighbourhood system N around 0 we can construct all other neighbourhood systems

In mammalian cells, vinculin is a membrane-cytoskeletal protein in focal adhesion plaques, involved in linkage of integrin adhesion molecules to the actin cytoskeleton. Vinculin is a cytoskeletal protein associated with cell-cell and cell-matrix junctions, where it is thought to function as one of several interacting proteins involved in anchoring F-actin to the membrane. Discovered independently by Benny Geiger and Keith Burridge, its sequence is 20%-30% similar to α-catenin, which serves a similar function. Binding alternately to talin or α-actinin, vinculin's shape and, as a consequence, its binding properties are changed; the vinculin gene occurs as a single copy and what appears to be no close relative to take over functions in its absence. Its splice variant metavinculin needs vinculin to heterodimerize and work in a dependent fashion. Vinculin is a 117-kDa cytoskeletal protein with 1066 amino acids; the protein contains an acidic N-terminal domain and a basic C-terminal domain separated by a proline-rich middle segment.

Vinculin consists of a globular head domain that contains binding sites for talin and α-actinin as well as a tyrosine phosphorylation site, while the tail region contains binding sites for F-actin and lipids. There is an 835 amino acid N-terminal head, split into four domains; this is linked to the C-terminal tail with a linker region. The recent discovery of the 3D structure sheds light on how this protein tailors its shape to perform a variety of functions. For example, vinculin is able to control the cell’s motility by altering its shape from active to inactive; when in its ‘inactive’ state, vinculin’s conformation is characterized by the interaction between its head and tail domains. And, when transforming to the ‘active’ form, such as when talin triggers binding, the intramolecular interaction between the tail and head is severed. In other words, when talin’s binding sites of α-helices bind to a helical bundle structure in vinculin’s head domain, the ‘helical bundle conversion’ is initiated, which leads to the reorganization of the α-helices, resulting in an new five-helical bundle structure.

This function extends to cancer cells, regulating their movement and proliferation of cancer to other parts of the body. Cell spreading and movement occur through the process of binding of cell surface integrin receptors to extracellular matrix adhesion molecules. Vinculin is associated with focal adhesion and adherens junctions, which are complexes that nucleate actin filaments and crosslinkers between the external medium, plasma membrane, actin cytoskeleton; the complex at the focal adhesions consists of several proteins such as vinculin, α-actinin and talin, at the intracellular face of the plasma membrane. In more specific terms, the amino-terminus of vinculin binds to talin, which, in turn, binds to β-integrins, the carboxy-terminus binds to actin and paxillin-forming homodimers; the binding of vinculin to talin and actin is regulated by polyphosphoinositides and inhibited by acidic phospholipids. The complex serves to anchor actin filaments to the membrane and thus, helps to reinforce force on talin within the focal adhesions.