## functional programming – My implementation of Clojure’s assoc-in

This is my implementation of Clojure’s `assoc-in` function. I am looking for tips on making it more idiomatic and in general, better.

``````(defn my-assoc-in
(m (& ks) v)
(let (sorted-keys (reverse ks))
(loop (curr-key (first sorted-keys)
rem-keys (rest sorted-keys)
curr-map (get-in m (reverse rem-keys))
acc-val  (assoc curr-map curr-key v))
(if (empty? rem-keys)
acc-val
(let (next-key (first rem-keys)
next-rem (rest rem-keys)
next-map (get-in m next-rem)
next-val (assoc next-map next-key acc-val))
(recur next-key next-rem next-map next-val))))))
``````

## Finding two nontrivial functional dependencies that follow from F (but are not in F)

Given below is the set F of functional dependencies for the relational schema:

``````R = {A,B,C,D,E,F,G}.

F = {AB → C,BC → D,G → F,AE → FG}
``````

Show two nontrivial functional dependencies that follow from F:

Which is :

AB → CD

ABC → D

I don’t know how those two derived from? Could someone walk me through? thanks

## functional analysis – Relation between fractional and integer Sobolev norms

I encountered a situation where I have to add two norms defined on the boundary:

$$C_1 ||u||^2_{L_2(partialOmega)} + C_2 ||u||^2_{H^{3/2}(partial Omega)},$$

but do not really know how to manipulate this expression. Is there a relation between the norms?

## Functional inequalities involving the condition \$left(int_0^t f(x)dxright)^2 ge int_0^t f(x)^3dx\$

I was reading the solution to a functional inequality in an article when the author made the following remark without giving any proof: let $$f(x): (0, infty)to(0, infty)$$ be locally integrable and such that
$$left(int_0^t f(x)dxright)^2 ge int_0^t f(x)^3dx$$
for all $$t>0$$. Then, the following statement is true:

$$int_0^t f(x)^gamma dx le frac{1}{gamma +1}left(2int_0^t f(x)dxright)^{(gamma + 1)/2}$$ for all positive $$t$$ and $$gamma in (1,3)$$.

Again, there is no proof in the article, so I don’t know if this is fairly easy to prove or very involved. One thing that might be worth mentioning is that the inequalities above become exact when $$f(x)=x$$. I am wondering if anyone has an idea or have seen these before.

## functional analysis – On a subspace that is isomorphic to a dense subspace

Let $$X$$ be an infinite dimensional Banach space and let $$M$$ be a dense subspace of $$X$$, i.e., $$overline{M}=X$$. Let $$N$$ be another subspace of $$X$$ such that $$N$$ is topologically isomorphic to $$M$$.
Then, is it true that $$N$$ is necessarily dense in $$X$$, i.e., $$overline{N}=X$$ ? I think (intuitively) this should be true. But, how to proceed to prove this fact? Could anybody provide a hint?

## functional analysis – Biconjugate formula in non-separated locally convex spaces

I have a locally convex space $$X$$ with topological dual $$X^*$$ and coupling $$langle x,x^* rangle:=x^*(x), xin X, x^*in X^*$$.

For $$f:Xtooverline{mathbb{R}}$$ one defines its convex conjugate by $$f^*(x^*):=sup{langle x,x^* rangle-f(x)mid xin X}$$ and similarly for $$g:X^*tooverline{mathbb{R}}$$ one defines its convex conjugate by $$g^*(x):=sup{langle x,x^* rangle-g(x^*)mid x^*in X^*}$$.

The biconjugate formula states that for a proper convex lower semicontinuous $$f:Xtooverline{mathbb{R}}$$ one has $$f^{**}=f$$.

The proof we went over in class is based on $$X$$ being Hausdorff separated and relies on separation theorems for the epigraph.

My question is whether the biconjugate formula holds when the space $$X$$ is NOT Hausdorff separated?

## functional analysis – Show that a sequence is eventually null

Let H be a separable Hilbert space and $${e_n}$$ and $${f_n}$$ be orthonormal basis in H. Define the linear operator:

$$begin{equation*} Sx = sum_{n=1}^{infty} mu_n (x,e_n)f_n. end{equation*}$$

I have to prove that if the dimension of Im S is finite, then $${mu_n}$$ is eventually null.

This is my attempt.

Let’s suppose that $$rank; T=k$$.

$$Sx= sum_{n=1}^infty alpha_n(x|e_n)f_n= sum_{n=1}^k alpha_n(x|e_n)f_n+ sum_{n=k+1}^infty alpha_n(x|e_n)f_n= sum_{n=1}^k alpha_n(x|e_n)f_n$$

So we have:

$$begin{equation} label{0} sum_{n=k+1}^infty alpha_n(x|e_n)f_n=0 end{equation}$$

As $${f_n}$$ is a basis:
$$alpha_n(x|e_n) =0 quad forall ngeq k+1$$

I don’t know how to conclude rigorously that $$alpha_n=0$$.

I’d appreciate if someone could revise this.

Thank you.

## Functional analysis: how to find a sequence that converges with the fixed random variable

Suppose $$mathscr {A}$$ is a convex set $$L ^ infty (P)$$, $$mathscr {B}$$ is the closing of $$mathscr {A}$$ under the topology $$σ (L ^ infty (P), L ^ 1 (P))$$. For each $$X in mathscr {B}$$, we can find a sequence $${X_n } _ {n geq 1} subseteq mathscr {A}$$ S t. $$X_n xrightarrow {L_1} X$$?

I thought about the problem a lot but still had no answer, so I ask for help here. Can you give me proof of existence or a counterexample? Thank you.

## Functional analysis: example of a non-reflective Banach space and two sequences.

Leave $$(E, mathcal {A}, mu)$$ be a space of finite measure and $$X$$ To be a Banach space. The set of all Bochner integrable functions from $$E$$ inside $$X$$ is denoted by $$mathcal {L} _X ^ 1$$.

Yes $$X$$ is thoughtful, we have the following theorem

Theorem 1

Leave $$(f_n) _ {n geq 1} subset mathcal {L} _ {X} ^ 1$$ it is a sequence with: $$sup_n int_ {E} { | f_n | d mu} < infty.$$
Then there is $$h _ { infty} in mathcal {L} _ { mathbb {R}} ^ 1$$ and a subsequence $$(g_k) _k$$ of $$(f_n) _n$$ such that for each subsequence $$(h_m) _m$$ of $$(g_k) _k$$ : $$frac {1} {i} sum_ {j = 1} ^ {i} {h_j (t)} to h _ { infty} (t) text {weakly in} X text {a.e. }$$

Proof of this result exists in the article "Infinite dimension extension of a Komlos theorem"by Erik J. Balder (theorem A)

Yes $$X$$ not thoughtful, we have the following theorem

Theorem 2

Leave $$(f_n) _ {n geq 1} subset mathcal {L} _ {X} ^ 1$$ it is a sequence with:
$$begin {cases} bullet ~~ {f_n (t) } text {is relatively weakly compact a.e.,} \ bullet ~~ sup_n int_ {E} { | f_n | d mu} < infty. \ end {cases}$$
Then there is $$h _ { infty} in mathcal {L} _ { mathbb {R}} ^ 1$$ and a subsequence $$(g_k) _k$$ of $$(f_n) _n$$ such that for each subsequence $$(h_m) _m$$ of $$(g_k) _k$$ : $$frac {1} {i} sum_ {j = 1} ^ {i} {h_j (t)} to h _ { infty} (t) text {weakly in} X text {a.e. }$$

Proof of this result exists in the article "Infinite dimension extension of a Komlos theorem"by Erik J. Balder (theorem B)

My problem:

I want an example of a non-reflective space of Banach and two sequences, so that:

• The first sequence is limited in $$mathcal {L} _ {X} ^ 1$$ but it does not verify the consequence of theorem 1.

• The second sequence that verifies the hypotheses of Theorem 2 and its consequences.