If series converges absolutely with respect to some norm, then it also converges absolutely with respect to any kind of equivalent norm.
I need to prove this assertion, but I have no idea from where to start.
2026-03-27 20:31:37.1774643497
Proof about converging absolutely with respect to equivalent norm
72 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in SEQUENCES-AND-SERIES
- How to show that $k < m_1+2$?
- 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
- Negative Countdown
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Show that the sequence is bounded below 3
- A particular exercise on convergence of recursive sequence
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Powers of a simple matrix and Catalan numbers
- Convergence of a rational sequence to a irrational limit
- studying the convergence of a series:
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 NORMED-SPACES
- How to prove the following equality with matrix norm?
- Closure and Subsets of Normed Vector Spaces
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Minimum of the 2-norm
- Show that $\Phi$ is a contraction with a maximum norm.
- Understanding the essential range
- Mean value theorem for functions from $\mathbb R^n \to \mathbb R^n$
- Metric on a linear space is induced by norm if and only if the metric is homogeneous and translation invariant
- Gradient of integral of vector norm
Related Questions in ABSOLUTE-CONVERGENCE
- Does one-sided derivative of real power series at edge of domain of convergence
- Every rearrangement of an absolutely convergent series converges to the same sum (Rudin)
- Prove $\int_{\pi}^{\infty}\frac{\cos(x)}{x}dx$ is convergent
- Conditional convergent improper Riemann integral vs. Lebesgue Integral
- Pointwise, uniform and absolute convergence of function-series $\sum_{n=1}^\infty f_n$ with $f_n=(-1)^n\frac{x}{n}$
- Does it make mathematical sense to do an absolute convergence test if the original series diverges?
- Prove absolute convergence given the following inequality
- Proving a space is not complete by finding an absolutely convergent series
- Convergence and absolute convergence of sums
- Absolute convergence of $\sum_{n=1}^{\infty}z_n^2$
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?
Let $\|\cdot \|_1$ and $\|\cdot \|_2$ be equivalent norms on a normed vector space $S.$ That is, the norms generate the same topology. Then the norms are uniformly equivalent : There exist positive $A, B$ such that for all $v\in S$ we have $A\|v\|_1\leq \|v\|_2\leq B\|v\|_1$.
Otherwise, with $(i,j)=(1,2)$ or $(i,j)=(2,1) ,$ there exists $T=\{v_n\}_{n\in N}\subset S$ with $\|v_n\|_i=1$ and $\|v_n\|_j<1/n$ for each $n\in N.$ With respect to the $j$-norm, the vector $0$ belongs to $\overline T,$ but with respect to the $i$-norm, $0\not \in \overline T,$ so the norms are not equivalent.
That should be enough to solve the problem.