Norms in Finite Dimensional Spaces

A set of vectors that is closed under addition and scalar multiplication, satisfying certain axioms.

A set of numbers where addition, subtraction, multiplication, and division (except by zero) are defined and follow certain rules.

Multiplying a vector by a number from the field K.

A function that measures the 'size' of a vector, satisfying nonnegativity, homogeneity, and the triangle inequality.

A seminorm that also satisfies the condition that a non-zero vector has a non-zero norm.

The norm of the sum of two vectors is less than or equal to the sum of their norms.

The norm of a scaled vector is equal to the absolute value of the scaling factor multiplied by the norm of the original vector.

The norm of any vector is always greater than or equal to zero.

To prove that a function f(t) is equal to zero for all t in the interval [a, b] given that the integral of its absolute value from a to b is zero, we need to show that if the integral vanishes, then the function itself must vanish everywhere within the interval.

The integral of a continuous and nonnegative function over an interval is zero if and only if the function itself is zero everywhere within the interval.

C^1(I) represents the space of all continuously differentiable functions f defined on a given interval I. These functions are differentiable, and their derivatives are also continuous.

This normed space consists of the function space C^1(I) with the norm k·k1,1 defined as the sum of the integral of the absolute value of the function and the integral of the absolute value of its derivative over the interval I.

This normed space comprises the function space C^1(I) equipped with the norm k·k1,∞ defined as the supremum of the sum of the absolute value of the function and its derivative taken over all points in the interval I.

A semi-norm is a function like a norm, but it may not satisfy the uniqueness property (i.e., it could be zero for non-zero functions). A true norm requires that the norm be zero only for the zero function.

The open ball centered at x0 with radius r in a normed space X is the set of all elements x in X whose distance to x0 is strictly less than r.

The closed ball centered at x0 with radius r in a normed space X is the set of all elements x in X whose distance to x0 is less than or equal to r.

A set where all elements are within a finite distance from a fixed point (usually the origin). In other words, all elements of the set are contained within a ball with a finite radius.

A set where every sequence within the set has a subsequence that converges to a point within the set.

A set that includes all its limit points. Meaning if a sequence within the set converges to a point, that point is also in the set.

A sequence in a normed space converges to a point if the distance between the sequence terms and the point approaches zero as the sequence index increases.

A sequence formed by taking some elements from a larger sequence, maintaining their order.

A set where elements are functions. The norm in a function space measures the 'size' of the function.

A mapping between two normed spaces, which preserves linearity and continuity. Meaning it keeps the scale of the input and ensures smooth transitions in outputs.

Lq(Ω, F, P) represents a space of functions defined on a measurable space (Ω, F) with a probability measure P, where the q-th power of the absolute value of the function is integrable with respect to the measure P. This essentially means functions whose q-th power has finite expected value.

kXkq represents the Lq-norm of a function X, which is defined as the q-th root of the integral of the q-th power of the absolute value of the function X with respect to the measure P.

A measure space is a mathematical construct consisting of a set (S), a sigma-algebra (Σ) of subsets of S, and a measure (µ) that assigns a non-negative value to each set in Σ. It provides a framework for measuring the 'size' or 'weight' of subsets of S.

The counting measure, denoted as µ, assigns a value equal to the number of elements in a set to each subset of N. For example, µ({1, 2, 3}) = 3.

`p(N) is a space of sequences of real numbers defined over the natural numbers (N). It contains all sequences (xn)n where the p-th power of their absolute value is summable.

kxkp represents the p-norm of a sequence x = (xn)n in p(N), calculated by taking the p-th root of the sum of the p-th powers of the absolute values of the sequence elements. It measures the 'size' or 'magnitude' of the sequence in p(N).

A normed space is a vector space equipped with a norm, which is a function that assigns a non-negative real number (called the 'norm') to each vector, representing its 'size' or 'magnitude'.

p(N) is a special case of Lp(S, µ) for a specific measure space (S, Σ, µ) where S = N, Σ = P(N), and µ is the counting measure. This means p(N) is effectively a space of sequences, where the 'size' of each sequence is measured using the p-norm.

A series ∑ xn in a normed space (X, ||·||) is convergent if there exists an element x in X such that the limit as N approaches infinity of the norm of the difference between the partial sum sN and x is equal to 0. This means the partial sums get arbitrarily close to x as N increases.

A series ∑ xn in a normed space (X, ||·||) is absolutely convergent if the sum of the norms of the individual terms ∑ ||xn|| is finite.

In a Banach space, every absolutely convergent series is also convergent. This means if the sum of the norms of the terms is finite, then the series itself converges to a point in the space.

Completeness. A Banach space is complete, which means every Cauchy sequence converges to a point in the space. When a series is absolutely convergent, its partial sums form a Cauchy sequence, and therefore must converge in the Banach space.

We can prove that the partial sums of the series (Sn) form a Cauchy sequence. Since the series is absolutely convergent, the sum of the norms of the terms is finite, allowing us to show that the distance between any two partial sums becomes arbitrarily small as N increases, satisfying the Cauchy criterion. Since the Banach space is complete, the Cauchy sequence of partial sums converges to a point in the space.

The proof utilizes the normalized vector of a non-zero vector x in Rd, which allows the relationship between norms in Rd and the sphere to be established.

Proposition 1.5.2 states that all norms in a finite dimensional vector space are equivalent. This means there are constant factors relating any two norms, implying they measure 'size' in a similar fashion.

Two norms N1 and N2 are related by constant factors, meaning you can bound one norm by another multiplied by these constants. This emphasizes their equivalence.

Φ is a bijection (one-to-one and onto) mapping between the finite dimensional vector space (X, k · k) and Rd, establishing a continuous correspondence between them.

Continuity of Φ indicates that small changes in x in the space (X, k · k) lead to small changes in Φ(x) in Rd. It preserves the notion of closeness between elements.

Φ being bijective ensures that each element in (X, k · k) has a unique corresponding element in Rd and vice-versa. This establishes a one-to-one correspondence between the spaces.

Both concepts contribute to the idea of 'similarity' between spaces. The equivalence of norms establishes a relationship between how size is measured, while Φ's continuity maintains this relationship across the mapping.

A continuous bijection Φ allows us to treat a finite dimensional space (X, k · k) and Rd as essentially the same, even though they might be different sets. It enables transferring properties and results between them.