What is the difference between norm and Seminorm?

In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.

What is quotient space in vector space?

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by “collapsing” N to zero. The space obtained is called a quotient space and is denoted V/N (read “V mod N” or “V by N”).

What is dimension of quotient space?

Definition 1.4 (Quotient Space). If M is a subspace of a vector space X, then the quotient space X/M is X/M = {f + M : f ∈ X}. Since two cosets of M are either identical or disjoint, the quotient space X/M is the set of all the distinct cosets of M.

What is the meaning of quotient topology?

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the …

Is a vector space a topological space?

A topological vector space is a vector space (an algebraic structure) which is also a topological space, this implies that vector space operations are continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

Is every metric is a pseudo metric?

Any metric space is a pseudometric space.

Why is it called quotient space?

Of course, the word “divide” is in quotation marks because we can’t really divide vector spaces in the usual sense of division, but there is still an analog of division we can construct. This leads the notion of what’s called a quotient vector space.

Is quotient map Open map?

The map f : X → Y is a closed map if for each closed set A ⊆ X the set f(A) is closed in Y . Note. If p : X → Y is continuous and surjective and p is either open or closed, then p is a quotient map. However, there are quotient maps that are neither open nor closed (see Munkres Exercise 22.3).

Is Banach space a topological space?

Motivation. Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces. There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. A topological field is a topological vector space over each of its subfields.

What is not a topological space?

A vector space is in turn not a topological space unless you define a topology on it. The comment from Jacky is explaining that given any vector space, you could for example give it the discrete topology, thus giving you a vector space which is also a topological space.