## norms – What are the functions such that \$ lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p\$?

Let $$1 leq p leq 2$$. I am looking for a characterization of the couples $$(f,g)$$ of functions $$f,g in L_p(mathbb{R})$$ such that
$$lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p.$$

For $$p = 2$$, this relation is satisfied if and only if $$langle f, g rangle = 0$$. For $$p = 1$$, it has been shown in this post that the condition is equivalent to $$f g geq 0$$ almost everywhere.
For a general $$p$$, the relation is clearly satisfied as soon as the product $$fg=0$$ almost everywhere. Is this latter sufficient condition also necessary?

If not, then what if we reinforce the condition with
$$lVert alpha f + beta g rVert_p^p = lVert alpha f rVert_p^p + lVert beta g rVert_p^p$$
for any $$alpha, beta in mathbb{R}$$?

## formal languages – Using pumping lemma to prove that \$L = { a^ib^j mid lvert i – j rvert le 2 } \$ is irregular

Given the following language:

$$L = { a^ib^j mid lvert i – j rvert le 2 }$$

I am trying to prove that it is not regular. On the one hand my intuition tells me that the language is non-regular as there is no way of tracking $$a^{i’s}$$ and $$b^{j’s}$$.

However, when I try to prove that it is irregular using the pumping lemma I have trouble finding which word I should use to arrive at a contradiction.

Any suggestions?

## functional analysis – If \$f_nrightarrow f\$ in \$L^2\$ and \$lVert f_nrVert_inftyle C\$ then why is \$fin L^infty\$?

This exercise from the $$L^p$$-theory section of my measure theory course is giving me some trouble.

Let $$(f_n)_nsubset L^2cap L^infty$$ be a sequence such that $$f_nrightarrow fin L^2$$ in $$L^2$$ and $$lVert f_nrVert_inftyle C$$ for all $$n$$. I want to show that

1. $$fin L^infty$$ and
2. $$f_nrightarrow f$$ in $$L^p$$ for any $$2le p

My most promising attempt at 1. was
$$lVert frVert_infty=lVert f-f_n+f_nrVert_inftyle lVert f-f_nrVert_infty+C,$$
but how can we bound $$lVert f-f_nrVert_infty$$?

I’m having similar problems with 2., except of course there we want to bound $$lVert f-f_nrVert_p$$.

## If \$d\$ is a metric in \$X\$ and \$A subset X\$, does \$lvert d(x,A) – d(y,A) rvert le d(x,y) phantom{5} forall x, y in X\$?

I´m trying to prove that if $$d$$ is a metric in $$X$$ and $$A subset X$$, then:

$$lvert d(x,A) – d(y,A) rvert le d(x,y), phantom{5} forall x, y in X$$

The question seems very simple but I´m having problems to solve it. Any suggestions?

## complex analysis – Analytic continuation of a function on the right half-plane to a region enclosing the circle \${ lvert z rvert = 3}\$

I am trying to solve the following question:

Show that there exists an analytic function $$f$$ in the open right half-plane such that $$(f(z))^2 + 2f(z) equiv z^2$$. Show that your function $$f$$ can be continued analytically to a region
containing the set $${z in mathbb{C} , lvert z rvert = 3}$$.

I can solve the first part. Since $$z^2+1$$ is a non-vanishing holomorphic function in the open right half-plane, which is simply connected, the function admits an analytic square root $$h(z)$$ in this region; it then suffices to choose $$h(z)-1$$ as our required function $$f(z)$$.

I don’t know how to proceed with the analytic continuation part though. Any suggestions?

## Functional analysis – Uniform binding to \$ lVert chi _ { {u_n = 0 }} rVert_ {W ^ {s, p} ( Omega)} \$ for a limited sequence \$ u_n \$ in \$ H ^ 1_0 ( Omega) \$?

Suppose I have a sequence $$u_n to u$$ in the $$H ^ 1_0 ( Omega)$$ on a smooth and limited domain. For some $$p> 1$$ and $$s in (0, frac 12)$$it is possible to estimate the norm of the characteristic function of the zero level set $$u_n$$, $$lVert chi _ { {u_n = 0 }} rVert_ {W ^ {s, p} ( Omega)}$$
in relation to norms of $$u_n$$? In particular, I am looking for a uniform that is bound for the expression above.

We know that it belongs, for example $$W ^ { epsilon, 2} ( Omega)$$ to the $$epsilon < frac 12$$I just want to know if it can be consistently limited.

## Why is \$ lvert e ^ {i * Im (s) * log (n)} rvert = 1 \$

Where $$n in mathbb {N}$$ and $$s in mathbb {C}$$,

## For which \$ alpha geq 1 \$ is \$ left lVert x right rVert ^ alpha \$ differentiable?

To let $$left lVert cdot right rVert_ infty$$ to be a norm $$mathbb {R} ^ n$$,

How can you find out for whom? $$alpha geq 1$$ the picture $$f$$ With $$f (x): = left lVert x right rVert ^ alpha$$ is completely differentiable in $$0$$?

## Why are \$(lVert .rVert_p) \$-norms equivalent in infinite direct sum?

If for i$$in mathbb{N}$$ the space $$X_i$$ is a Banach space, Why are the spaces
$$(oplus _{i=1}^{infty} X_i)_{lVert . rVert_p}$$ for
$$1leq p$$ equivalent?