Intereting Posts

if $2f(x)+f''(x)=-xf'(x)$ show that $f(x)$and $f'(x)$ are bounded on $R$
Godel's pairing function and proving c = c*c for aleph cardinals
Every subspace of the dual of a finite-dimensional vector space is an annihilator
How is arccos derived?
Do dynamic programming and greedy algorithms solve the same type of problems?
Integrate $ \int_0^\infty ( x A+I)^{-1} A -I \frac{1}{c+x} dx $ where $A$ is psd and $c>0$
$ \lim_{x \rightarrow \ + \infty}(\sqrt{x^2 + 2x} – \sqrt{x^2 – 7x})$
If the size of 2 subgroups of G are coprime then why is their intersection is trivial?
Which groups have precisely two automorphisms
Rings and modules of finite order
Topology on cartesian product and product topology.
Inducing homomorphisms on localizations of rings/modules
On the definition of projective vector bundle.
Extensions: Spectrum
Why is quantified propositional logic not part of first-order logic?

Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well.

If I knew that $B(Y)$ is closed I’d have a bounded linear map $B^{-1}:B(Y)\rightarrow Y$ by the bounded inverse theorem. Therefore $A=B^{-1}BA$ is bounded. How to prove the claim if $B(Y)$ is not closed.

- Linear combinations of delta measures
- How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
- Weak limit and strong limit
- Rearrangement of series in Banach space and absolute convergence
- An application of Riesz' Lemma
- How to prove this inequality in Banach space?

- $\sigma(x)$ has no hole in the algebra of polynomials
- Confused about a version of Schauder's fixed point theorem
- An exercise about the definition of nuclear maps
- In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.
- Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$.
- Weak topologies and weak convergence - Looking for feedbacks
- If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)
- The Duals of $l^\infty$ and $L^{\infty}$
- Are there relations between elements of $L^p$ spaces?
- If $\|e_1+e_2+\cdots+e_n\|\leq C$ for all $n$ then $(e_i)_{i=1}^n$ is uniformly equivalent to the basis of $\ell_\infty^n$?

This is extended version of Nate Eldredge’s hint:

- Take $\{x_n\}$ such that $\lim_{n\to\infty}x_n= x$,$\lim_{n\to\infty}A(x_n)= y$.
- Show that $\lim_{n\to\infty}B(A(x_n))= B(A(x))$.
- Recall that $B$ is injective.
- Apply closed graph theorem.

- Why does this method for solving equations with complex number roots not always work?
- If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$
- Matrix linear algebra generators
- Have any discrete-time continuous-state Markov processes been studied?
- Multi-index power series
- Concrete description of (co)limits in elementary toposes via internal language?
- How can I write an equation that matches any sequence?
- Making the water gallon brainteaser rigorous
- Prove by induction. How do I prove $\sum\limits_{i=0}^n2^i=2^{n+1}-1$ with induction?
- Let $L_p$ be the complete, separable space with $p>0$.
- Is the following set compact
- how to show convergence in probability imply convergence a.s. in this case?
- Difference between “space” and “algebraic structure”
- Contour integral of $\int_0^{2\pi} \frac{1}{A – cos \theta} d\theta$
- Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?