Let $X$ be a Hilbert space. Let $\Lambda :X\to X$ be given by $$\Lambda g=f_1(g)+f_2(g)+f_3(g),$$ where $f_i:X\to X$. What I need to prove is that $\Lambda$ is a contraction. For some reason, I need to consider $$\|\Lambda g_1-\Lambda g_2\|^2_X=\|(f_1(g_1)-f_1(g_2))+(f_2(g_1)-f_2(g_2))+(f_3(g_1)-f_3(g_2))\|_X^2.$$ The latter equality is very unconvinient for me, because of short multiplication formulas. Instead of this, can I understand a norm in a different way and consider $$\|\Lambda g_1-\Lambda g_2\|^2_X=\|(f_1(g_1)-f_1(g_2))\|_X^2+\|(f_2(g_1)-f_2(g_2))\|_X^2+\|(f_3(g_1)-f_3(g_2))\|_X^2?$$
2026-03-25 03:07:25.1774408045
A problem with defining a norm to get a contraction
29 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 FIXED-POINT-THEOREMS
- Newton's method with no real roots
- Determine $ \ a_{\max} \ $ and $ \ a_{\min} \ $ so that the above difference equation is well-defined.
- Banach and Caristi fixed point theorems
- Show that $\Phi$ is a contraction with a maximum norm.
- Using Fixed point iteration to find sum of a Serias
- Map a closed function $f: (1,4) \rightarrow (1,4)$ without fixed point
- Stop criterium for fixed point methods
- Approximate solutions to nonlinear differential equations using an integral sequence
- Inverse function theorem via degree theory
- Fixed point of a map $\mathbb R^n \rightarrow \mathbb R^n$
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?
The desired equality will be in general false. True if the three differences are mutually orthogonal. See Pythagorean Theorem in Hilbert space. The usual procedure in this situation is using the triangule inequality.