Intereting Posts

The “set” of equivalence classes of things.
Topology on the space of universally integrable functions
If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?
How do I prove that a finite covering space of a compact space is compact?
Monic (epi) natural transformations
How to find all roots of the quintic using the Bring radical
Maximum of $E$ for $Z$ standard normal and $X$ independent of $Z$, two-valued, with $E=c$
What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$?
Winning strategies in multidimensional tic-tac-toe
Question about generating function in an article
A and B disjoint, A compact, and B closed implies there is positive distance between both sets
Degeneracy in Linear Programming
“Standard” ways of telling if an irreducible quartic polynomial has Galois group C_4?
finitely generated k-algebra and polynomial ring
How can $y$ and $y'$ be independent in variational calculus?

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.

- A compact operator is completely continuous.
- $C()$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$
- Cancellation law for Minkowski sums
- Is $W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$ complete?
- The group of invertible linear operators on a Banach space
- Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

- Bochner Integral vs. Riemann Integral
- Inclusion of $\mathbb{L}^p$ spaces, reloaded
- Is there a nonnormal operator with spectrum strictly continuous?
- A Riesz-type norm-preserving and bijective mapping between a Banach space and its dual
- Exercise books in functional analysis
- A counter example of best approximation
- space of bounded measurable functions
- Open Mapping Theorem: counterexample
- Positivity of the Coulomb energy in 2d
- Why is this inclusion of dual of Banach spaces wrong?

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.

- Suppose that G is a finite, nonabelian group with odd order. Show s is surjective, and hence bijective
- Is the sequence $a_{n}=\prod\limits_{i=1}^{n}\left(1+\frac{i}{n^2}\right)$ decreasing?
- A space $X$ is locally connected if and only if every component of every open set of $X$ is open?
- How do I find a Bezier curve that goes through a series of points?
- Why are addition and multiplication included in the signature of first-order Peano arithmetic?
- A niggling problem about frontier in topology
- Limit of a monotonically increasing sequence and decreasing sequence
- The number of grid points near a circle.
- Construct a real function with is exactly $C^2$ such that its first derivative does not vanish everywhere
- The homotopy category of complexes
- Can someone show me a proof of the general solution for 2nd order homogenous linear differential equations?
- Log Sine: $\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta.$
- How are lopsided binomials (eg $\binom{n}{n+1})?$ defined?
- Riemann zeta function and Bernoulli function
- Intuition behind the Axiom of Choice