* If is the norm ( usually noted as ) defined in the square-summable sequence space ℓ < sup > 2 </ sup > ( which also matches the usual distance in a continuous and isotropic cartesian space ), then

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

Coefficients ( a < sub > 0 </ sub >, a < sub > 1 </ sub >, b < sub > 1 </ sub >, a < sub > 2 </ sub >, b < sub > 2 </ sub >, ...) are in fact an element of an infinite-dimensional vector space ℓ < sup > 2 </ sup >, and thus Fourier series is a linear operator.

* A Hilbert space is separable if and only if it has a countable orthonormal basis, it follows that any separable, infinite-dimensional Hilbert space is isometric to ℓ < sup > 2 </ sup >.

For example, consider the right shift operator R on the Hilbert space ℓ < sup > 2 </ sup >,

Any such sequence belongs to the Hilbert space ℓ < sub > 2 </ sub >, so the Hilbert cube inherits a metric from there.

* The Erdős space ℓ < sup > p </ sup >( Z )∩ is a totally disconnected space that does not have dimension zero.

The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: on both the strength p of its poles, and on the vector ℓ separating them.

More precisely, for a given orbital momentum quantum number ℓ ( representing the azimuthal quantum number associated with angular momentum ), there are 2ℓ + 1 integral magnetic quantum numbers m ranging from-ℓ to ℓ, which restrict the fraction of the total angular momentum along the quantization axis so that they are limited to the values m. This phenomenon is known as space quantization.

The ba space of the power set of the natural numbers, ba ( 2 < sup > N </ sup >), is often denoted as simply and is isomorphic to the dual space of the ℓ < sup >∞</ sup > space.

Let S be the shift operator on the sequence space ℓ < sup >∞</ sup >( Z ), which is defined by ( Sx )< sub > i </ sub > = x < sub > i + 1 </ sub > for all x ∈ ℓ < sup >∞</ sup >( Z ), and let u ∈ ℓ < sup >∞</ sup >( Z ) be the constant sequence u < sub > i </ sub > = 1 for all i ∈ Z.

The " space " character had to come before graphics to make sorting easier, so it became position 20 < sub > hex </ sub >; for the same reason, many special signs commonly used as separators were placed before digits.

Several unusual applications, such as a nuclear battery or fuel for space ships with nuclear propulsion, have been proposed for the isotope < sup > 242m </ sup > Am, but they are as yet hindered by the scarcity and high price of this nuclear isomer.

By sweeping this surface through R < sup > 3 </ sup > as a function of the ion sequence input data, such as via ion-ordering, a volume is generated onto which positions the 2D detector positions can be computed and placed three-dimensional space.

At ambient conditions, berkelium assumes its most stable α form which has a hexagonal symmetry, space group P6 < sub > 3 </ sub >/ mmc, lattice parameters of 341 pm and 1107 pm.

So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.

Similarly, as an infinite-dimensional example, the Lebesgue space L < sup > p </ sup > is always a Banach space but is only a Hilbert space when p = 2.

However, it is expensive to grow and wastes space proportional to 2 < sup > h </ sup >-n for a tree of depth h with n nodes.

Just as kets and bras can be transformed into each other ( making into ) the element from the dual space corresponding with is where A < sup >†</ sup > denotes the Hermitian conjugate ( or adjoint ) of the operator A.

Four points P < sub > 0 </ sub >, P < sub > 1 </ sub >, P < sub > 2 </ sub > and P < sub > 3 </ sub > in the plane or in higher-dimensional space define a cubic Bézier curve.

* Take the Banach space R < sup > n </ sup > ( or C < sup > n </ sup >) with norm || x ||

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

Equipped with the topology of pointwise convergence on A ( i. e., the topology induced by the weak -* topology of A < sup >∗</ sup >), the character space, Δ ( A ), is a Hausdorff compact space.

ft ) of on-site storage space, and 9, 400 m < sup > 2 </ sup > ( 101, 000 sq.

T denotes the tube axis, and a < sub > 1 </ sub > and a < sub > 2 </ sub > are the unit vectors of graphene in real space.

The authors set about answering this fundamental question through a detailed investigation of the patient's ability, tactually, ( 1 ) to perceive figure and ( 2 ) to locate objects in space, with his eyes closed ( or turned away from the object concerned ).

* 1962 – The Mariner 2 unmanned space mission is launched to Venus by NASA.

Geometrically, one studies the Euclidean plane ( 2 dimensions ) and Euclidean space ( 3 dimensions ).

The Urban Land Institute ( ULI ) awarded the Battery Park City Master Plan its 2010 Heritage Award, for having " facilitated the private development of 9. 3 million square feet of commercial space, 7. 2 million square feet of residential space, and nearly 36 acres of open space in lower Manhattan, becoming a model for successful large-scale planning efforts and marking a positive shift away from the urban renewal mindset of the time.

Not every compact space is sequentially compact ; an example is given by 2 < sup ></ sup >, with the product topology.

The probability measure on Cantor space, sometimes called the fair-coin measure, is defined so that for any binary string x the set of sequences that begin with x has measure 2 < sup >-| x |</ sup >.

Chloroplast ultrastructure: 1. outer membrane 2. intermembrane space 3. inner membrane ( 1 + 2 + 3: envelope ) 4. stroma ( aqueous fluid ) 5. thylakoid lumen ( inside of thylakoid ) 6. thylakoid membrane 7. granum ( stack of thylakoids ) 8. thylakoid ( lamella ) 9. starch 10. ribosome 11. plastidial DNA 12. plastoglobule ( drop of lipids )

Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.

Since the map d restricts to 0 on I < sub > x </ sub >< sup > 2 </ sup > ( the reader should verify this ), d descends to a map from I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup > to the dual of the tangent space, ( T < sub > x </ sub > M )< sup >*</ sup >.

A complete installation will take in excess of 11. 5 GB of hard disk space, but usable configurations may require as little as 1 or 2 GB.

* Quantity (< span lang = grc > poson </ span >, how much ), discrete or continuous — examples: two cubits long, number, space, ( length of ) time.

For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:

Each of these strings p < sub > i </ sub > determines a subset S < sub > i </ sub > of Cantor space ; the set S < sub > i </ sub > contains all sequences in cantor space that begin with p < sub > i </ sub >.

The utility of Cauchy sequences lies in the fact that in a complete metric space ( one where all such sequences are known to converge to a limit ), the criterion for convergence depends only on the terms of the sequence itself.

The Cantor space is the collection of all infinite sequences of 0s and 1s.

This implies that for each natural number n, the set of sequences f in Cantor space such that f ( n )

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

But " having distance 0 " is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M. The original space is embedded in this space via the identification of an element x of M with the equivalence class of sequences converging to x ( i. e., the equivalence class containing the sequence with constant value x ).

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.

However, the number of possible distinct prion strains is likely far smaller than the number of possible DNA sequences, so evolution takes place within a limited space.

Sequence families are often determined by sequence clustering, and structural genomics projects aim to produce a set of representative structures to cover the sequence space of possible non-redundant sequences.

A first countable, separable Hausdorff space ( in particular, a separable metric space ) has at most the continuum cardinality c. In such a space, closure is determined by limits of sequences and any sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X.