## Functional analysis of fa – Numeric range of the tensor product of two matrices

Leave $$T in M_n$$. Is the following true?
$$bigcap limits_ {B in M_2 \ text {tr} (B) = 0} left {X in M_2: W (X) subseteq W (T) text {y} W ( B otimes X) subseteq W (B otimes T) right } subseteq bigcap limits_ {B in M_2} ​​ left {X in M_2: W (B otimes X) subseteq W (B otimes T) right }?$$ where $$W (S): = { langle Sx, x rangle: Vert x Vert = 1 }$$ it's called numerical range of $$S$$.

Comments: I have checked the above in MATLAB using some particular option of $$T, B$$ which is affirmative for those cases. So, I have tried to prove the above as follows:

• We know $$W (X) subseteq W (T)$$ if a map $$varphi: text {span} {I, T, T ^ * } rightarrow text {span} {I, X, X ^ * }$$ S t. $$varphi (aI + bT + cT ^ *) = aI + bX + cX ^ *$$ where $$a, b, c in mathbb {C}$$ is positive. So, by hypothesis, we have a positive map that says $$psi: text {span} {I_2 otimes I, text {span} {B _ { circ}, B _ { circ} ^ * } otimes text {span} {T , T ^ * } } rightarrow M_4$$ S t.$$psi (I_2 otimes I + B otimes T + B ^ * otimes T ^ *) = I_2 otimes I_2 + B otimes X + B ^ * otimes X ^ *$$ it's positive where $$B in M_2$$ S t $$tr (B) = 0$$ Y $$B _ circ} begin {pmatrix} 0 and 1 \ 0 and 0 \ end {pmatrix}.$$ Now, I have no idea if the map $$varphi$$ can extend positively to $$text {span} {I, B _ { circ}, B _ { circ} ^ * } otimes text {span} {I, T, T ^ * }$$ to get the required result.

It may be false, but I still don't have any counterexample. Thanks in advance. Any comments are greatly appreciated.

## Functional analysis: are there no true functions with real values ​​of two or more variables?

I see the following theorem somewhere, without complete proof.

For each irrational number $$lambda$$, there are continuous functions $$phi_k: (0,1) to mathbb R, k = 1,2,3,4,5$$, so that for all continuous functions $$f: (0,1) ^ 2 to mathbb R$$, there is a continuous function $$g: (0,1) a mathbb R$$such that $$f (x, y) = sum_ {k = 1} ^ 5 g ( phi_k (x) + lambda phi_k (y))$$.

There are some mysterious things about this result. Where does the number come from? $$5$$ come and why $$lambda notin mathbb Q$$?

Could anyone offer a reference for proof of this?

## Functional analysis of fa – Higher order functional derivatives

Leave $$E, F$$ Be Banach spaces. A continuous bilinear function $$langle cdot, cdot rangle: E times F to mathbb {R}$$ is named $$E$$-do-degenerate yes $$langle x, y rangle = 0$$ for all $$and in F$$ it implies $$x = 0$$ (Similarly for $$F$$-no-degenerate). Equivalently, the two maps of $$E$$ to $$F *$$ Y $$F$$ to $$E *$$ defined by $$x mapsto langle x, cdot rangle$$ Y $$y mapsto langle cdot, and rangle$$, respectively, are one to one. If they are isomorphisms (*), $$langle cdot, cdot rangle$$ is named $$E$$ or $$F$$-Strongly not degenerate. We say that $$E$$ Y $$F$$ they are in duality if there is a non-degenerated bilinear function $$langle cdot, cdot rangle: E times F to mathbb {R}$$, also called pairing from $$E$$ with $$F$$. If the functional is strongly non-degenerated, we say that the duality is strong.

Consider the following definition.

Definition: Leave $$E$$ Y $$F$$ be regulated spaces and $$langle cdot, cdot rangle$$ a $$E$$– non degenerate matching. Leave $$f: F to mathbb {R}$$ be Fréchet differentiable at the point $$alpha in F$$ (denote this derivative as $$Df (α)$$) The functional derivative $$delta f / delta alpha$$ from $$f$$ with respect to $$alpha$$ It is the unique element in $$E$$, if it exists, so that:
$$begin {eqnarray} Df ( alpha) ( gamma) = langle frac { delta f} { delta alpha}, gamma rangle tag {1} label {1} end {eqnarray}$$
for all $$gamma in F$$.

Now, I would like to know how to define higher order derivatives of functional derivatives. In other words, suppose Fréchet's derivative of $$f$$ to $$alpha$$, $$Df (α)$$ is Fréchet differentiable in $$beta in F$$. Is it possible to define $$delta2 f / delta beta delta alpha$$?

## calculation and analysis: possible error related to the BesselI derivative

In Mathematica 12.0, I execute the following code:

``````f(x_) = BesselI(0, x);
f'(x)
``````

that comes back `BesselI(1, x)` as expected. But if I enter

``````f(x_) = BesselI(0, 1.0 x);
f'(x)
``````

I get

``````0.5 (BesselI(1, 1. x) + BesselI(1, 1. System`Private`DerivativeX(1.)))
``````

Further, `D(f(x),x)` Returns the expected result. I tried to exit the kernel without changes. Is this a mistake or something is wrong with my installation?

## real analysis – Search: \$ lim limits_ {n to + infty} frac { left (1+ frac {1} {n ^ 3} right) ^ {n ^ 4}} { left ( 1+ frac {1} {(n + 1) ^ 3} right) ^ {(n + 1) ^ 4}} \$

Find :

$$lim limits_ {n to + infty} frac { left (1+ frac {1} {n ^ 3} right) ^ {n ^ 4}} { left (1+ frac {1} {(n + 1) ^ 3} right) ^ {(n + 1) ^ 4}}$$

My attempt: I don't know if it's right or not!
I use this rule:

$$lim limits_ {n to + infty} (f (x)) ^ {g (x)} = 1 ^ { infty}$$

Then :

$$lim limits_ {n to + infty} (f (x)) ^ {g (x)} = lim limits_ {n to + infty} e ^ {g (x) (f ( x) -1)}$$

So :

$$lim limits_ {n to + infty} frac { left (1+ frac {1} {n ^ 3} right) ^ {n ^ 4}} { left (1+ frac {1} {(n + 1) ^ 3} right) ^ {(n + 1) ^ 4}}$$

$$= lim limits_ {n to + infty} frac {e ^ { frac {n ^ {4}} {n ^ {3}}}} {e ^ { frac {(n + 1 ) 4} {(n + 1) 3}}$$

$$= lim limits_ {n to + infty} frac {e ^ {n}} {e ^ {n + 1}}$$

$$= frac {1} {e}$$

Is my approach wrong?

Is this called partial limit calculation?

## Functional analysis – How to calculate the Fréchet derivative of the \$ varepsilon function: H_ {per} 1 ([0,L]) times L_ {by} 2 ([0,L]) longrightarrow mathbb {R} \$

Yes $$varepsilon: H_ {per} ^ ((0, L)) times L_ {per} ^ {((0, L)) longrightarrow mathbb {R}$$ is given by
$$varepsilon (u, v) = frac {1} {2} int_ {0} ^ {L} big ({u_x} ^ 2 + v ^ 2 + 2 cos (u) big) ; dx, : forall : (u, v) in H_ {per} ^ {1} ((0, L)) times L_ {per} ^ 2 ((0, L)),$$
where $$L> 0$$ is the period of all functions of $$L_ {per} 2 ((0, L))$$ Y $$H_ {per} 1 ((0, L)) subset L_ {per} 2 ((0, L))$$. So what is the derivative of Fréchet or Gateaux of $$varepsilon$$? And how to calculate?

## Stochastic analysis: proof that a simulated random space from an SDE is a Hilbert space L2

I have an infinite set of simulated values ​​using a stochastic differential equation (similar to Ornstein-Uhlenbeck, but it can be any SDE). I need to use these values ​​to forecast future values ​​(similar to Longtsaff-Schwartz)

I want to show that these generated values ​​belong to an L2 space, so I can make an orthogonal projection (that is, regression).

Question:

1- I don't know where to start to show that some values ​​I got from a simulation are of bounded variance. Any pointer? book? video?

2- Now we apply some continuous functions to these values. Are they still Hilbert?

In formal terms:

I have:

$$mathbb {E} left ( Delta_x ( omega, t, t + delta) ^ 2 mid mathcal {F} _t right)$$

where $$Delta_x$$ is a $$mathcal {F} _t$$adapted process calculated as the difference between two values ​​of the simulated vectors over time (I have $$n$$ values ​​of these):

$${ Delta ^ 2 ( omega, t, t + delta)} _ n = {V ( omega, t + delta)} _ n- {V ( omega, t)} _ n$$

The norms that I want to use for the projection of these random variables are:

1. Simple Euclidean norm (to make a polynomial regression)
2. Weighted standard (to do Kernel regression):

$$langle Weight. ( Delta ^ 2 (.) – X_t), X_t) rangle$$

I need to show that the simulated variables belong to L2 (under these two standards) in order to justify an orthopedic projection.

Thank you!

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## Uniform Convergence in Analysis – Mathematics Stack Exchange

Consider the series 1 + x + x2 + x3 + …. We know that this series converges promptly to 1 / (1 – x) for x ∈ (−1,1). Show that
(a) Convergence is not uniform in (−1,1).
(b) The convergence is uniform in each closed and bounded interval (−a, a) where 0 <a <1. Show this using both the definition of uniform convergence and the Weierstrass-M test.

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