ChatGPT, answers 2022-12-19, MRENTHUSIASM

HUMAN:
Poignant, short, truthful, exact answers.
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Given that:
We are talking about modern science up to 2020 and try to teach people well versed in the field do keep explanations short to suit the target audience. Which are PhD holders in the field.

Question:
What is functional analysis?

ChatGPT:
[207: "functional analysis"] Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.), generally coming from some kind of limit procedure (e.g. Riemann integral, Cesàro mean, etc.), together with the linear operators defined on these spaces and having some kind of continuity property (e.g. bounded, compact, etc.).

Functional analysis has its origins in the study of spaces of functions and the formulation of mathematical analysis in terms of vector spaces, as well as in the study of differential and integral equations. It has applications in many fields, including physics, engineering, economics, and statistics.

Some of the key concepts in functional analysis include:

    Normed vector spaces: These are vector spaces equipped with a notion of "length" or "size" for vectors, called a norm. Normed vector spaces are often used to model infinite-dimensional spaces, such as the space of continuous functions on a compact interval, or the space of square-integrable functions on a measure space.

    Banach spaces: These are complete normed vector spaces, meaning that every Cauchy sequence in the space converges to a unique limit. Banach spaces are often used as the domain or codomain of linear operators, because the completeness property allows us to define the inverse of a bounded linear operator.

    Hilbert spaces: These are complete inner product spaces, meaning that they are equipped with an inner product that allows us to define the notion of "angle" between vectors and to use the Pythagorean theorem to compute vector lengths. Hilbert spaces are often used to model physical systems that have an underlying structure of symmetry or orthogonality.

    Continuous linear operators: These are linear operators that map one Banach space to another and have a certain degree of "continuity" with respect to the norm on the domain and codomain spaces. Continuous linear operators play a central role in the study of differential and integral equations, as well as in the study of quantum systems.

    Compact linear operators: These are continuous linear operators that have the additional property of "compactness", which means that they map bounded sets to relatively compact sets. Compact linear operators have many useful properties, including the fact that they have a discrete spectrum (i.e. only a finite number of eigenvalues).

Functional analysis is a rich and active field of research, with many deep connections to other areas of mathematics and physics. It is also a fundamental tool in the study of partial differential equations, operator theory, and harmonic analysis.

HUMAN:
Question:
What is a mathematical norm in that context? Do add visualization and three examples.

ChatGPT:
[232: "mathematical norm visualization examples"] A norm is a function that assigns a nonnegative "length" or "size" to each vector in a vector space. Formally, a norm is a function ||x|| that satisfies the following three properties:

    Positivity: ||x|| >= 0 for all x, and ||x|| = 0 if and only if x = 0.

    Homogeneity: ||ax|| = |a|*||x|| for all a in the field over which the vector space is defined (e.g. a could be a real number or a complex number) and all x in the vector space.

    Triangle inequality: ||x + y|| <= ||x|| + ||y|| for all x and y in the vector space.

A normed vector space is a vector space equipped with a norm. Normed vector spaces are often used to model infinite-dimensional spaces, such as the space of continuous functions on a compact interval, or the space of square-integrable functions on a measure space.

Here are three examples of norms:

    The Euclidean norm: This norm is defined on R^n (the space of n-tuples of real numbers) as ||x|| = sqrt(sum(x_i^2)). Geometrically, this norm corresponds to the Euclidean distance from the origin to the point represented by the vector x.

[asy] /* Made by MRENTHUSIASM / size(200pt); import graph; defaultpen(linewidth(0.7pt)+fontsize(10pt)); real labelscalefactor = 0.5; / changes label-to-point distance / pen dps = linetype("4 4")+linewidth(0.7pt)+blue; pen dsh = linetype("6 6")+linewidth(0.7pt)+red; pen dots = black; / dots for grid / pair F = (1/sqrt(2),1/sqrt(2)); / F is foot of the perpendicular from origin to x axis */ pair O = (0,0), X = (1,0), Y = (0,1); draw((-1.5,0)--(1.5,0),dots,EndArrow(5)); draw((0,-1.5)--(0,1.5),dots,EndArrow(5)); label("$x_1$",X,SW); label("$x_2$",Y,W); pair A = (3/sqrt(2),0), B = (0,3/sqrt(2)); pair C = (A+B)/2; draw(O--A--B--cycle,linewidth(1.5pt)+blue+fontsize(8pt)); draw(O--C--B,dsh); label("$||x||$",(O+C)/2,SW,fontsize(8pt)); draw(rightanglemark(O,F,A,2)); draw(rightanglemark(O,F,B,2)); pair[] ps = {O,A,B}; dot(ps); label("$O$",O,SW); label("$A$",A,SE); label("$B$",B,N); [/asy]

    The l^p

HUMAN:
Continue, you stopped in the middle of your output.

ChatGPT:
I apologize for the abrupt ending. Here is the remainder of my response:

    The l^p norm: This norm is defined on R^n as ||x||_p = (sum(|x_i|^p))^(1/p). The value of p can be any positive real number. When p = 1, this norm is called the l^1 norm, and it is sometimes referred to as the "taxicab" or "Manhattan" norm because it measures the distance that a taxicab would need to travel to visit all the coordinates in the vector. When p = 2, this norm is called the l^2 norm, and it is sometimes referred to as the "Euclidean" norm because it is the square root of the sum of the squares of the coordinates. When p = infinity, this norm is called the l^infinity norm, and it is the maximum absolute value of the coordinates.

[asy] /* Made by MRENTHUSIASM / size(200pt); import graph; defaultpen(linewidth(0.7pt)+fontsize(10pt)); real labelscalefactor = 0.5; / changes label-to-point distance / pen dps = linetype("4 4")+linewidth(0.7pt)+blue; pen dsh = linetype("6 6")+linewidth(0.7pt)+red; pen dots = black; / dots for grid / pair F = (1/sqrt(2),1/sqrt(2)); / F is foot of the perpendicular from origin to x axis */ pair O = (0,0), X = (1,0), Y = (0,1); draw((-1.5,0)--(1.5,0),dots,EndArrow(5)); draw((0,-1.5)--(0,1.5),dots,EndArrow(5)); label("$x_1$",X,SW); label("$x_2$",Y,W); pair A = (1,1), B = (2,2), C = (A+B)/2; draw(O--A--B--cycle,linewidth(1.5pt)+blue+fontsize(8pt)); draw(O--C--B,dsh); label("$||x||_1$",(O+C)/2,SW,fontsize(8pt)); draw(rightanglemark(O,F,A,2)); draw(rightanglemark(O,F,B,2)); pair[] ps = {O,A,B}; dot(ps); label("$O$",O,SW); label("$A$",A,NE); label("$B$",B,NE); [/asy]

    The l^2 norm: This norm is defined on R^n as ||x||_2 = sqrt(sum(x_i^2)). Geometrically, this norm corresponds to the Euclidean distance from the origin to the point represented by the vector x. It is often used as the default norm in machine learning and optimization problems because it has a number of useful properties, such as the fact that it is differentiable almost everywhere and that it is invariant under rotations.

[asy] /* Made by MRENTHUSIASM */ size(200pt); import graph; defaultpen(