What is B-metric space?

What is B-metric space?

Definition 1 Let X be a set and let d be a function from X × X into [0, ∞). Then (X, d) is said to be a b-metric space if the following hold: d(x, y) = 0 ⇔ x = y; d(x, y) = d(y, x) (symmetry); There exists K ≥ 1 satisfying d(x, z) ≤ K(d(x, y) + d(y, z)) for any x, y, z ∈ X (K-relaxed triangle inequality).

Who introduced B-metric space?

1. Introduction. The idea of b-metric was initiated from the works of Bourbaki [1] and Bakhtin [2]. Czerwik [3] gave an axiom which was weaker than the triangular inequality and formally defined a b-metric space with a view of generalizing the Banach contraction mapping theorem.

What is D in metric space?

A metric space is a set X together with a function d (called a metric or “distance function”) which assigns a real number d(x, y) to every pair x, y. X satisfying the properties (or axioms): d(x, y)

What is a subspace of a metric space?

A subset M⊂X is called a subspace of X, written (M,d)⊂(X,d), if M is endowed with the same metric as X, called the induced metric on M. Subspaces of metric spaces are themselves metric spaces. Ex. Any set becomes a metric space when endowed with the discrete metric d(x,y):={1,x≠y,0,x=y.

What is l2 In metric space?

The space l2(N) is one of metric spaces containing convergent sequences. 2.5. Space l2(P) Kadak, et al [1] define a distance function pq induced by the metric dp defined in Equation 1. Proposition 2.4 A distance function pq : lq(P) × lq(P) → R+ ∪ {0} for 1 ≤ q < ∞ defined.

Is r/d complete?

Example: consider the metric defined by d(x,y)=|x3−y3|. Let f:ℝ⟶ℝ be the injective function defined by f(x)=x3. The image of f is ℝ which is closed, so (ℝ,d) is complete. On the other hand if d(x,y)=|arctanx−arctany|, Im(f)=]−π/2;π/2[ is not closed, so (ℝ,d) is not complete.

What is a product metric space?

In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces. which metrizes the product topology. The most prominent product metrics are the p product metrics for a fixed. : It is defined as the p norm of the n-vector of the distances measured in n subspaces: For.

Is every vector space a metric space?

No, not necessarily. A vector space with no additional structure has no metric, and is thus not a metric space. You can give a vector space more structure so that it is also a metric space.

What is L2 function space?

On a measure space , the set of square integrable L2-functions is an -space. Taken together with the L2-inner product with respect to a measure , (1) the -space forms a Hilbert space.

Is L2 a compact?

Later in this lecture we will show that the closed unit ball in the sequence spaces ℓ∞, c0, ℓ1 and ℓ2 is not compact, and we will give examples of compact sets in these spaces.

Is R n complete?

Definition Rn is a complete metric space. Every Cauchy sequence in Rn converges to a point of Rn.

Is RA metric space?

A metric space is a set X together with such a metric. The prototype: The set of real numbers R with the metric d(x, y) = |x – y|. This is what is called the usual metric on R.