How to show $d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$ is a metric?

$d_1(x_1,y_1)$ and $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about proving the triangle inequality for $d$ ? Thank you for your help. Answer Consider the function $f(a,b) = \sqrt{a^2 + b^2}$ on $[0,\infty)\times [0,\infty)$. Then … Read more

Characteristic functions dense in simple functions in L1L^1?

Consider L1(X,Y) where X=Y=[0,1] and ⟨f,g⟩=∫fg. Is the set of characteristic functions {χA} dense in the set of simple functions {s}, in the sense there is a sequence {χAn} ⟨χAn,g⟩→⟨s,g⟩ for any continuous g∈L∞(X,Y)? I read this answers here ( but I didn’t fully understand. It seems the answer is yes? Hope anyone offer some … Read more

Is the sequence of functions, fn(x)=xnf_n(x) = x^n Cauchy in C([0,1])C([0,1])? Is it Cauchy in L2([0,1])L^2([0,1])?

Is the sequence of functions, fn(x)=xn Cauchy in C([0,1])? Is it Cauchy in L2([0,1])? I know that it is not Cauchy in the space of continuous functions. I know that the L-space is complete and suspect that the sequence will be Cauchy in that space then. My idea to prove it is by contradiction. I … Read more

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I approach this? I have proved that the function is a metric already. But I can’t think of how elements are mapped for some … Read more

Prove that $T(B)$ is relatively compact in $C([a,b])$.

Let $B$ be the unit ball in $C([a,b])$. Define for $f\in C([a,b])$, $$Tf(x)=\int_a^b (-x^2+e^{-x^2+y})f(y)dy.$$ Prove that $T(B)$ is relatively compact in $C([a,b])$. My attempt: If $|f(x)| \le 1$ on $[a,b]$. Then $T(f)$ is bounded. We only need to show that $T(f)$ is equi-continuous by Arzela-Ascoli theorem. $T(f)(x)-T(f)(z)=\int_a^b (x^2+z^2)f(y)dy+\int_a^b (e^{-x^2}-e^{-z^2})e^y f(y)dy$. Then I don’t know how … Read more

A Banach space of (Hamel) dimension κ\kappa exists if and only if κℵ0=κ\kappa^{\aleph_0}=\kappa

A Banach space of (Hamel) dimension κ exists if and only if κℵ0=κ. How will we prove the converse implication. One sided implication for Hilbert Space is proved in question: Can you equip every vector space with a Hilbert space structure? If we don’t assume Axiom of Choice, and we have a Banach space with … Read more

If a normal element of a C* algebra has real spectrum, then it is self-adjoint

Let A be a C∗-algebra. Suppose x is a normal element of A and spect(x) lies in R. Prove that x is self-adjoint. I tried the following: using spect(x)=¯spect(x∗) conclude that λ−a=λ−a∗⟹a=a∗ for λ in spect(a). Is this valid? Answer There may well be a simpler solution. (But see a certain comment below.) Since a … Read more

Trace evaluation via complex analysis

We are given $U$, $V$ unitary matrices of size $N \times N$ whose spectral decomposition is known (in my specific problem, $N=4$, and $U$, $V$ are matrices with real coefficients but we can keep it general). We are asked to evaluate $$ \mathrm{Tr} \left( U^{m_1}V^{k_1} \cdots U^{m_n}V^{k_n}\right) $$ for $m_1,\ldots,m_n$ and $k_1,\ldots,k_n$ given integers $\geq … Read more