On a finite-dimensional vector space,
all norms are equivalent
but this is not true for infinite dimensional vector spaces. All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same).
Is every normed space is an inner product space?
The abstract spaces—metric spaces, normed spaces, and inner product spaces—are all examples of what are more generally called “topological spaces.” These spaces have been given in order of increasing structure. That is, every inner product space is
a normed space
, and in turn, every normed space is a metric space.
Are all vector spaces inner product spaces?
Even if a (real or complex) vector space admits an inner product (e.g. finite dimensional ones), a vector space need
not come with an inner product
. An inner product is additional structure and it is often useful and enlightening to see what does and what does not require the additional structure of an inner product.
Does every finite dimensional vector space have an inner product?
Every finite dimensional vector space can made into an inner product space
with the same dimension
.
Are all vector spaces fields?
Every field is a vector space
but not every vectorspace is a field. I need an example for which a vector space is also a field.
Which space is not inner product space?
The
vector space R over Q
is not an inner product space. This is a simple answer.
How do you find an orthonormal basis?
First, if we can find an orthogonal basis, we
can always divide each of the basis vectors by their magnitudes to arrive at an orthonormal basis
. Hence we have reduced the problem to finding an orthogonal basis. Here is how to find an orthogonal basis T = {v
1
, v
2
, … , v
n
} given any basis S.
Is norm a metric space?
All norms are metrics
, and normed spaces (vector spaces with a norm) have a lot more structure than general metric spaces. Anything that holds in a metric space will also hold for a normed space.
Is every normed space is complete?
Every normed space can
be isometrically embedded onto
a dense vector subspace of some Banach space, where this Banach space is called a completion of the normed space. This Hausdorff completion is unique up to isometric isomorphism.
What is the norm of a vector space?
The norm is a function,
defined on a vector space, that associates to each vector a measure of its length
. In abstract vector spaces, it generalizes the notion of length of a vector in Euclidean spaces.
What is an F vector space?
A vector space over F — a.k.a. an F-space — is
a set (often denoted V ) which has a binary operation +V (vector addition) defined on it
, and an operation ·F,V (scalar multiplication) defined from F × V to V . (So for any v, w ∈ V , v +V w is in V , and for any α ∈ F and v ∈ V α·F,V v ∈ V .
Can vector space empty?
Vector spaces need a zero vector (an additive identity) just like groups need an identity element. So
empty sets cannot be vector spaces
.
How do you prove a vector space is finite dimensional if it has?
For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is finite-dimensional
if the dimension of V is finite
, and infinite-dimensional if its dimension is infinite.
Which one is not vector space?
Similarily, a vector space needs to allow any scalar multiplication, including negative scalings, so
the first quadrant of the plane (even including the coordinate axes and the origin)
is not a vector space.
Is R 2 a vector space?
The vector space R2 is
represented by the usual xy plane
. Each vector v in R2 has two components. The word “space” asks us to think of all those vectors—the whole plane. Each vector gives the x and y coordinates of a point in the plane : v D .
Is a line a vector space?
A line through the origin is
a one-dimensional vector space
(or a one-dimensional vector subspace of R2). A plane in 3D is a two-dimensional subspace of R3. The vector space consisting of zero alone is a zero dimensional vector space.
