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Let $ W $ be a sub vector space of $ mathbb{R}^n $. How can we determine if $ W $ admits an integer basis?

This is equivalent to asking how to determine if $ W cap mathbb{Z}^n $ spans $ W $.

Let $G$ be a locally compact Polish (or compact) group acting continuously on a locally compact Polish (or compact) space $X$, and $mu$ a Borel measure on $X$. To be sure, continuity of the action means that the map $(g, x) in G times X mapsto g cdot x in X$ is continuous with respect to the product topology on $G times X$. Let $X/G$ denote the orbit space endowed with the quotient topology, and $pi : X rightarrow X/G$ denote the orbit map. A cross-section to the orbit map is a map $s: X/G rightarrow X$ satisfying $s circ pi = 1_{X}$. If $s$ and $t$ are measurable or continuous cross-sections to the orbit map, then it is known that their images $s(X/G)$ and $t(X/G)$ are measurable (closed in the case of a continuous cross-section with compact $G$ and $X$). What is the relationship between $mu(s(X/G)$ and $mu(t(X/G))$? Is it reasonable to expect $mu(s(X/G)) = mu(t(X/G))$?

PS: It is enough for me to consider the case of $X = G$, that is, $X$ is the underlying topological space of $G$, and the action of conjugation, and $mu$ Haar measure on $G$. Thanks.

Consider $R^3$ as an inner product space in relation to scalarmultiplication. Find all vectors in the subspace

$$Spanbigg(bigg[begin{matrix}1\2\1
end{matrix}bigg],bigg[begin{matrix}3\4\1
end{matrix}bigg]bigg)subseteq R^3,$$

which are orthogonal to the vector $bigg[begin{matrix}-1\1\1
end{matrix}bigg]$

Any help will be appreciated. I tried finding the random vector $v=bigg[begin{matrix}x_1\x_2\x_3
end{matrix}bigg]$ in the plane which i found to be ${x_3cdotbigg[begin{matrix}1\-1\1
end{matrix}bigg]bigg}$ but i dont know where to go from here.

The starting point was that $ Gamma'(x+1)=Gamma(x+1)psi(x+1)$ where $psi(x+1)=-gamma+H_{x}$ . Hence $$ [x!]’ = x!biggl[-gamma+sum_{k=1}^{x}frac{1}{k}biggl]$$ For example $ [4!]’ = 24[-gamma+1+1/2+1/3+1/4]$ which gives 36.1462 that, put in the tangent equation, gives us exactly the tangent for x=4! Let me know if I made any mistakes in the derivation/generalization!

Let $W_2(mu,nu)$ denote the $2$-Wasserstein distance between two given probability measures $mu$ and $nu$ on $mathbb R^n$. For a probability measure $mu$ and $f:mathbb R^nto mathbb R^n$, let $f_{#}mu=mucirc f^{-1}$ denote the push-forward of $mu$ under $f$, i.e. $(f_{#}mu)(B)=mu(f^{-1}(B))$ for every Borel set $B$ in $mathbb R^n$. Why does the following inequality hold true?
$$W^2_2(f_{#}mu,g_{#}mu)leq int_{mathbb R^n}|f(x)-g(x)|^2,dmu(x) $$
for all $mu$-measurable functions $f,g:mathbb R^ntomathbb R^n$.

Some comment: the product measure $f_{#}muotimes g_{#}mu$ is a so-called transport plan and by definition of the Wasserstein distance
$$W^2_2(f_{#}mu,g_{#}mu)leq int_{mathbb R^ntimes mathbb R^n}|x-y|^2,d(f_{#}muotimes g_{#}mu)(x,y)=int_{mathbb R^ntimes mathbb R^n}|f(x)-g(y)|^2,dmu(x),dmu(y).$$