4
votes
1answer
75 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
1
vote
0answers
39 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
3
votes
1answer
79 views

Are $L_p$ spaces of functions with separable support separable?

Let $X$ be a separable space. Is $L_p$$(X, \mu, V)$ a separable space? Here, $(V, |\cdot|_V)$ is a normed space. And a norm of $L_p(X, \mu, V)$ is: $$ \|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p ...
3
votes
1answer
100 views

Weak* continuity

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $\mathcal{M}^+$ the set of nonnegative Radon measures on $B$ and $\mathcal{M}^2$ the set of $\mathbb{R}^2 \text{-valued}$ Radon measures on $B.$ I ...
1
vote
2answers
34 views

Another question about integrable functions with a transform

I am an engineering student, and taking a real analysis course at demand of my advisor, my inexperience in proofs is giving me hard time. I stumbled upon this example, whose proof left as an exercise. ...
3
votes
0answers
91 views

Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
4
votes
1answer
120 views

Convergence in $L_\infty$ and $L_1$ even if infinite measure space

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. In the literature, assuming the measure space $X$ has finite measure, if $f_n$ converges to ...
3
votes
1answer
89 views

What are the $n$-th degree minimal polynomials for $L^p([-1,1])$?

It is known (even by me) that the Chebyshev polynomial of degree $n$ (of the first kind) is the minimal polynomial in the space $L^{\infty}([-1,1])$ for a fixed $n$ and leading coefficient $2^n$. ...
3
votes
2answers
263 views

Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...