# on 06-Apr-2018 (Fri)

#### Annotation 1729541311756

 #topology In geometry, an affine transformation, affine map[1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Annotation 1729651674380

 #topology In geometry, an affine transformation, affine map[1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes.

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In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, tho

#### Original toplevel document

Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Flashcard 1729653247244

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#topology
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In geometry, an affine transformation preserves [...objects...].
points, straight lines and planes

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In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes.

#### Original toplevel document

Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Annotation 1729654820108

 #topology An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

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map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. <span>An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. <span><body><html>

#### Original toplevel document

Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Flashcard 1729656392972

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#topology
Question
affine transformation does not necessarily preserve [...] between lines
angles

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#### Parent (intermediate) annotation

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An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

#### Original toplevel document

Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Flashcard 1729657965836

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#topology
Question
affine transformation does not necessarily preserve [...] between points
distances

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An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

#### Original toplevel document

Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Flashcard 1729659538700

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#topology
Question
An affine transformation preserve [...] between points lying on a straight line.
ratios of distances

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#### Parent (intermediate) annotation

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An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

#### Original toplevel document

Affine transformation - Wikipedia
s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#### Annotation 1731726806284

 #stochastics The Wiener process is characterised by the following properties:[1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance , has continuous paths: With probability , is continuous in ...

Wiener process - Wikipedia
Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

#### Flashcard 1731730476300

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#stochastics
Question

The Wiener process starts at [...

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The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance ,

#### Original toplevel document

Wiener process - Wikipedia
Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

#### Flashcard 1731732835596

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#stochastics
Question

The Wiener process has [...] increments: for every the future increments , are independent of the past values ,

independent

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The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance , has continuous paths: With

#### Original toplevel document

Wiener process - Wikipedia
Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

#### Flashcard 1731734408460

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#stochastics
Question

The Wiener process has independent increments: for every [...

the future increments , are independent of the past values ,

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The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance , has continuous paths: With probability , is continuous in .

#### Original toplevel document

Wiener process - Wikipedia
Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

#### Flashcard 1731736767756

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#stochastics
Question

The Wiener process has [...]: is normally distributed with mean and variance ,

Gaussian increments

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ad> The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance , has continuous paths: With probability , is continuous in . <html>

#### Original toplevel document

Wiener process - Wikipedia
Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

#### Flashcard 1731738340620

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#stochastics
Question

in the Wiener process , is normally distributed with [...

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r process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with <span>mean and variance , has continuous paths: With probability , is continuous in . <span><body><html>

#### Original toplevel document

Wiener process - Wikipedia
Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

#### Annotation 1737411661068

 #python Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets:

The Goodies
use you can’t put a print statement inside the loop. I suggest that you use them only if the computation is simple enough that you are likely to get it right the first time. And for beginners that means never. 19.3 Generator expressions <span>Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets: >>> g = (x**2 for x in range(5)) >>> g at 0x7f4c45a786c0> The result is a generator object that knows how to iterate through a sequence of values. But unlike a

#### Flashcard 1737415331084

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#python
Question
[...] are similar to list comprehensions, but with parentheses instead of square brackets:
Generator expressions

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Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets:

#### Original toplevel document

The Goodies
use you can’t put a print statement inside the loop. I suggest that you use them only if the computation is simple enough that you are likely to get it right the first time. And for beginners that means never. 19.3 Generator expressions <span>Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets: >>> g = (x**2 for x in range(5)) >>> g at 0x7f4c45a786c0> The result is a generator object that knows how to iterate through a sequence of values. But unlike a

#### Flashcard 1737416903948

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#python
Question
Generator expressions are similar to [...], but with parentheses instead of square brackets:
list comprehensions

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#### Parent (intermediate) annotation

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Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets:

#### Original toplevel document

The Goodies
use you can’t put a print statement inside the loop. I suggest that you use them only if the computation is simple enough that you are likely to get it right the first time. And for beginners that means never. 19.3 Generator expressions <span>Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets: >>> g = (x**2 for x in range(5)) >>> g at 0x7f4c45a786c0> The result is a generator object that knows how to iterate through a sequence of values. But unlike a

#### Annotation 1796927786252

 #hilbert-space An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

Hilbert space - Wikipedia
x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

#### Annotation 1796930407692

 #hilbert-space The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

Hilbert space - Wikipedia
to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

#### Annotation 1796932504844

 #hilbert-space A second example is the space ℂ2 whose elements are pairs of complex numbers z = (z1, z2) . Then the inner product of z with another such vector w = (w1,w2) is given by The real part of 〈z,w〉 is then the four-dimensional Euclidean dot product.

Hilbert space - Wikipedia
w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z

#### Annotation 1796936699148

 #hilbert-space The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted || x || , and to the angle θ between two vectors x and y by means of the formula

Hilbert space - Wikipedia
t product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. <span>The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos ⁡ θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacem

#### Annotation 1796940893452

 #hilbert-space the completeness of Euclidean space means that a series that converges absolutely also converges in the ordinary sense.

Hilbert space - Wikipedia
→ 0 as N → ∞ . {\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.} <span>This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square

#### Flashcard 1796944563468

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#hilbert-space
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the Euclidean space is equipped with [...operation...].

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the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product.

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Hilbert space - Wikipedia
ble spaces 9 Orthogonal complements and projections 10 Spectral theory 11 See also 12 Remarks 13 Notes 14 References 15 External links Definition and illustration[edit source] Motivating example: Euclidean space[edit source] One of <span>the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x · y. If x and y are represented in Cartesian coordinates, then the dot product is defined by

#### Flashcard 1798939479308

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#hilbert-space
Question
An operation on pairs of vectors, that satisfies [...], is known as a (real) inner product.
the three properties of the dot product

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An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

#### Original toplevel document

Hilbert space - Wikipedia
x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

#### Flashcard 1798941052172

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#hilbert-space
Question
An operation on [...], that satisfies the three properties of the dot product, is known as a (real) inner product.
pairs of vectors

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An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

#### Original toplevel document

Hilbert space - Wikipedia
x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

#### Flashcard 1798942625036

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#hilbert-space
Question
An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a [...]
(real) inner product.

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An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

#### Original toplevel document

Hilbert space - Wikipedia
x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

#### Flashcard 1798944984332

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#hilbert-space
Question

The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both [...] and [...]

length and angle

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The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted || x || , and to the angle θ between two vectors x and y by means of the formula

#### Original toplevel document

Hilbert space - Wikipedia
t product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. <span>The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos ⁡ θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacem

#### Flashcard 1798947343628

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#hilbert-space
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[...] relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.
Multivariable calculus in Euclidean space

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Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.

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Hilbert space - Wikipedia
\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange). <span>Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞

#### Flashcard 1798949702924

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#hilbert-space
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Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that [...]
limits exist.

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Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.

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Hilbert space - Wikipedia
\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange). <span>Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞

#### Flashcard 1798952062220

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#hilbert-space
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the [...] of Euclidean space means that a series that converges absolutely also converges in the ordinary sense.

completeness

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the completeness of Euclidean space means that a series that converges absolutely also converges in the ordinary sense.

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Hilbert space - Wikipedia
→ 0 as N → ∞ . {\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.} <span>This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square

#### Flashcard 1798954421516

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#hilbert-space
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The complex plane ℂ is equipped with a notion of [...]

magnitude

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The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

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Hilbert space - Wikipedia
to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

#### Flashcard 1798955994380

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#hilbert-space
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The notion of magnitude in the complex plane is [...]

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The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

#### Original toplevel document

Hilbert space - Wikipedia
to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

#### Flashcard 1798957567244

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#hilbert-space
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the complex modulus is defined as [...description...]

the square root of the product of z with its complex conjugate

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The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

#### Original toplevel document

Hilbert space - Wikipedia
to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

#### Flashcard 1798959140108

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#hilbert-space
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the complex modulus | z | is defined as [...formula...]

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The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

#### Original toplevel document

Hilbert space - Wikipedia
to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

#### Flashcard 1798964645132

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#hilbert-space
Question

the elements of space ℂ2 are [...]

pairs of complex numbers

z = (z1, z2)

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A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ) . Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by The real part of 〈z,w〉 is then the four-dimensional Euclidean dot product. </sp

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Hilbert space - Wikipedia
w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z

#### Flashcard 1798967004428

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#hilbert-space
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the inner product of two vectors (z and w) in the space ℂ2 is given by [...]

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A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ) . Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by The real part of 〈z,w〉 is then the four-dimensional Euclidean dot product.

#### Original toplevel document

Hilbert space - Wikipedia
w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z

#### Flashcard 1798969363724

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#hilbert-space
Question

The real part of the inner product〈z,w〉in the space ℂ2 is the [...] in the Euclidean space