Let's assume that we have a unital Banach algebra $T$ and we define sine and cosine using the normal power series definition as for $\mathbb{R}$. Does the Pythagorean trigonometric identity still hold?
Let $(\mathcal X,\mathcal F)$ be a measurable space and $\mathcal P$ the set of probability measures on this space. With weak convergence, a basis of the corresponding weak topology is a collection of the following sets:
$$U(P,\alpha)=\left\{Q:\left|\int f_idP-\int f_idQ\right|<\alpha,i=1,\ldots...
Let $H$ be a Hilbert space and $N: D(N)\subset H \to H$ be some self-adjoint operator with domain $D(N)$ dense in $H$ and spectrum
$$\sigma(N) \subset (-\infty, \alpha]$$
for some fixed $\alpha <0.$ We assume further that
$$e^{tN}(D(N))\subset D(N), \quad \forall t\in (0, 1].$$
Let now $g: H ...
Let $H$ be a Hilbert space and $N: D(N)\subset H \to H$ be some self-adjoint operator with domain $D(N)$ dense in $H$ and spectrum
$$\sigma(N) \subset (-\infty, \alpha]$$
for some fixed $\alpha <0.$ We assume further that
$$e^{tN}(D(N))\subset D(N), \quad \forall t\in (0, 1].$$
Let now $g: H ...
