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Questions and Answers
What is the relation of all norms over a finite dimensional vector space X?
What is the relation of all norms over a finite dimensional vector space X?
The proposition states that the 1-norm is less than or equal to the norm N2 in every case.
The proposition states that the 1-norm is less than or equal to the norm N2 in every case.
False
What is the formula for the norm of a vector x in Rd as defined in the content?
What is the formula for the norm of a vector x in Rd as defined in the content?
kxk1 = Σ|xi| for i=1 to d
In the mapping Φ: X → Rd, x is represented as _____ in terms of the basis {e1,..., ed}.
In the mapping Φ: X → Rd, x is represented as _____ in terms of the basis {e1,..., ed}.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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What does the normalization of vector x refer to in the context?
What does the normalization of vector x refer to in the context?
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The mapping Φ is not continuous.
The mapping Φ is not continuous.
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Describe what is established when applying the previous proposition to (X, N1).
Describe what is established when applying the previous proposition to (X, N1).
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What is the supremum norm defined as in normed space X = C1([0, 1])?
What is the supremum norm defined as in normed space X = C1([0, 1])?
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If K is a compact subset of a normed space, then K must also be bounded.
If K is a compact subset of a normed space, then K must also be bounded.
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What does it mean for a subset K of a normed space to be compact?
What does it mean for a subset K of a normed space to be compact?
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A compact set is always _____.
A compact set is always _____.
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Match the following concepts with their definitions:
Match the following concepts with their definitions:
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Which of the following statements is true about the supremum of a sequence in a compact set?
Which of the following statements is true about the supremum of a sequence in a compact set?
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All closed sets in a normed space are compact.
All closed sets in a normed space are compact.
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Can you provide an example of a compact set in R?
Can you provide an example of a compact set in R?
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What characterizes a series in a normed space as being convergent?
What characterizes a series in a normed space as being convergent?
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In a Banach space, any absolutely convergent series is also convergent.
In a Banach space, any absolutely convergent series is also convergent.
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Define an absolutely convergent series.
Define an absolutely convergent series.
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The partial sums of the series are defined as sN = _____ for N ∈ N.
The partial sums of the series are defined as sN = _____ for N ∈ N.
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What is necessary to prove that a sequence is a Cauchy sequence in a Banach space?
What is necessary to prove that a sequence is a Cauchy sequence in a Banach space?
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Every sequence in a Banach space is automatically Cauchy.
Every sequence in a Banach space is automatically Cauchy.
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What does it mean for a Banach space to be complete?
What does it mean for a Banach space to be complete?
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Which of the following is NOT a property of a K-vector space?
Which of the following is NOT a property of a K-vector space?
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A seminorm on a vector space must ensure that N(x) is always greater than or equal to zero.
A seminorm on a vector space must ensure that N(x) is always greater than or equal to zero.
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What is the term for a function N: X → R that satisfies the properties of nonnegativity, homogeneity, and the triangle inequality?
What is the term for a function N: X → R that satisfies the properties of nonnegativity, homogeneity, and the triangle inequality?
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For any vectors x, y in a K-vector space, the property that states N(x+y) ≤ N(x) + N(y) is referred to as the __________ inequality.
For any vectors x, y in a K-vector space, the property that states N(x+y) ≤ N(x) + N(y) is referred to as the __________ inequality.
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Match the following properties with their definitions:
Match the following properties with their definitions:
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What is always true for a norm but may not be true for a seminorm?
What is always true for a norm but may not be true for a seminorm?
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What symbol represents the additive identity element in a K-vector space?
What symbol represents the additive identity element in a K-vector space?
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What is the relationship between the spaces Lq and `q?
What is the relationship between the spaces Lq and `q?
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The properties of scalar multiplication in a K-vector space include that 1x = x for all x in the space.
The properties of scalar multiplication in a K-vector space include that 1x = x for all x in the space.
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The sequence space `1 (N) is equivalent to the space L1 (S, µ) for some special choice of S and µ.
The sequence space `1 (N) is equivalent to the space L1 (S, µ) for some special choice of S and µ.
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What does the notation kXkq represent?
What does the notation kXkq represent?
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`p (N) is defined as the set of sequences x = (xn) such that the sum of |xn|^p is ___ for n from 1 to infinity.
`p (N) is defined as the set of sequences x = (xn) such that the sum of |xn|^p is ___ for n from 1 to infinity.
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If µ(S) < ∞, which measure space configuration is valid?
If µ(S) < ∞, which measure space configuration is valid?
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The counting measure can only be used for finite sets.
The counting measure can only be used for finite sets.
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In the definition of `p (N), the symbol p is required to be greater than ___ for the space to be normed.
In the definition of `p (N), the symbol p is required to be greater than ___ for the space to be normed.
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In the context of continuous functions, what does it mean if $\int_a^b |f(t)|dt = 0$?
In the context of continuous functions, what does it mean if $\int_a^b |f(t)|dt = 0$?
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The function space C(I) includes only functions that are differentiable but not necessarily continuously differentiable.
The function space C(I) includes only functions that are differentiable but not necessarily continuously differentiable.
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What is the definition of a normed space?
What is the definition of a normed space?
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The open ball centered at $x_0$ with radius $r$ is defined as $B(x_0, r) = { x \in X ; kx - x_0 k < r }$ while the closed ball is defined as $B_c(x_0, r) = { x \in X ; kx - x_0 k \underline{______} r }$
The open ball centered at $x_0$ with radius $r$ is defined as $B(x_0, r) = { x \in X ; kx - x_0 k < r }$ while the closed ball is defined as $B_c(x_0, r) = { x \in X ; kx - x_0 k \underline{______} r }$
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Match the following definitions with their corresponding terms:
Match the following definitions with their corresponding terms:
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Which of the following properties is NOT satisfied by a semi-norm?
Which of the following properties is NOT satisfied by a semi-norm?
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The integral of a continuous function is always nonnegative.
The integral of a continuous function is always nonnegative.
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What is the significance of the sup norm in function spaces?
What is the significance of the sup norm in function spaces?
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Study Notes
Lecture Notes for Master's Degree in Stochastics and Data Sciences
- Course covered: Analysis
- Academic Year: 2024-2025
- Instructor: Bertrand Lods
- Original lectures from 2015-2016, with updates for 2024-2025
Contents
- Normed spaces: the norm and its properties, topology of normed spaces, equivalent norms, product spaces, continuity of functions, fundamental examples like Lebesgue spaces (L¹), LP-spaces (1 ≤ p ≤ 8), space of linear applications, compactness, and finite-dimensional spaces
- Banach spaces: Cauchy sequences, completeness, examples, fundamental properties of Banach spaces, problems, complements involving compact operators
- Inner product spaces and Hilbert spaces: Definitions, general properties, Hilbert spaces, projection theorem, orthogonal complements, orthonormal families, Hilbert bases, and more properties of the dual space/Hahn-Banach Theorem
- Complements about Lebesgue spaces: Useful inequalities (Jensen, Minkowski), integral depending on a parameter, convolution
- Fourier analysis: Fourier transform in L¹(Rd), main properties, Riemann-Lebesgue theorem, inversion formula
- Laplace transform: definition, linearity, continuity, initial and final value theorems
Additional Information
- Proofs that students need to prepare and understand are indicated in the margins with a symbol.
- Lecture notes were created with support from the project: "Start up of the Master's Degree in Stochastics and Data Science."
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Description
Explore the relationships and properties of norms in finite dimensional vector spaces through this comprehensive quiz. Test your understanding of various norms, compactness, and continuity in normed spaces. The quiz includes matching terms, definitions, and true or false statements related to norms.
