Why are reflexive spaces important intuitively? I know that it brings about many good properties: Properties of reflexive Banach spaces. However, what is so special about reflexive spaces that make these good properties happen? Is there a general intuition of this?
2026-02-23 11:46:40.1771847200
Intuition on why Reflexive Spaces are Important
415 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in REAL-ANALYSIS
- how is my proof on equinumerous sets
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Is this relating to continuous functions conjecture correct?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Absolutely continuous functions are dense in $L^1$
- A particular exercise on convergence of recursive sequence
Related Questions in FUNCTIONAL-ANALYSIS
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- Prove or disprove the following inequality
- Unbounded linear operator, projection from graph not open
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- Elementary question on continuity and locally square integrability of a function
- Bijection between $\Delta(A)$ and $\mathrm{Max}(A)$
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- If $A$ generates the $C_0$-semigroup $\{T_t;t\ge0\}$, then $Au=f \Rightarrow u=-\int_0^\infty T_t f dt$?
Related Questions in HILBERT-SPACES
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- hyponormal operators
- a positive matrix of operators
- If $S=(S_1,S_2)$ hyponormal, why $S_1$ and $S_2$ are hyponormal?
- Is the cartesian product of two Hilbert spaces a Hilbert space?
- Show that $ Tf $ is continuous and measurable on a Hilbert space $H=L_2((0,\infty))$
- Kernel functions for vectors in discrete spaces
- The space $D(A^\infty)$
- Show that $Tf$ is well-defined and is continious
- construction of a sequence in a complex Hilbert space which fulfills some specific properties
Related Questions in INTUITION
- How to see line bundle on $\mathbb P^1$ intuitively?
- Intuition for $\int_Cz^ndz$ for $n=-1, n\neq -1$
- Intuition on Axiom of Completeness (Lower Bounds)
- What is the point of the maximum likelihood estimator?
- Why are functions of compact support so important?
- What is it, intuitively, that makes a structure "topological"?
- geometric view of similar vs congruent matrices
- Weighted average intuition
- a long but quite interesting adding and deleting balls problem
- What does it mean, intuitively, to have a differential form on a Manifold (example inside)
Related Questions in REFLEXIVE-SPACE
- Weak convergence in a reflexive, separable and infinite Banach space.
- Weak and Weak* convergences implying reflexivity
- Non-Reflexivity of $L^1$ by finding a particular operator in the dual space?
- My attempt to show the canonical embedding $c_0\rightarrow c_0^{**}$ is not surjective?
- Is the Unit ball of $X^{**} $ weak*- sequentially compact?
- How to get $\ell_1$ is not reflexive from $c_0^*$ is not reflexive?
- Example on $X^*$ is reflexive but $\sigma(X^*,X)\neq\sigma(X^*,X^{**})$ for a normed space $X$.
- A direct example on $X^*$ is reflexive but the weak topology and weak* topology on $X^*$ are not equal for a normed space $X$.
- How to show a normed space $X$ is reflexive if the weak topology and weak* topology on $X^*$ are equal?
- Bounded sequence has Cauchy subsequence w.r.t. $ (x|y)_0:=\sum^\infty_{n=1} 2^{-n}\phi_n(x)\phi_n(y).$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
I am not sure how much intuition there is, when it comes to Banach spaces and why we "like" certain properties more than others except for the reason you mentioned: if we take them away things get more complicated and I guess in some sense less intuitive and less desirable and more pathological.
I guess one way of looking at it would be to recognize that in some sense the "nicest" Banach spaces are the Hilbert spaces. They have many of the important properties we need in order to model and study more complex problems for which the finite dimensional case is simply not "rich enough". (Say more rich structures when it comes to spectra of linear operators and so forth.) However, they also do share a lot with their smaller finite dimensional brothers. They can be canonically identified with its dual, they admit a useful concept of a bases and they are indeed all reflexive.
So maybe one way of thinking about Banach spaces and trying to classify them is to see "how much they differ" from those nice Hilbert spaces. As it turns out non-reflexivity is one of those important properties in order to rescue for instance some topological intuition.
Not sure if that helped. Maybe more like a long comment.