# All Questions

262 views

### Prove a space has a countable dense subset

I'm doing this exercise in Munkres book and got no clue about the solution. Hope some one can help me solve this. Show that the product space $R^{I}$, where $I = [0,1]$, has a countable dense ...
280 views

### Polynomials $P(x)$ satisfying $P(2x-x^2) = (P(x))^2$

I am looking (to answer a question here) for all polynomials $P(x)$ satisfying the functional equation given in the title. It is not hard to notice (given that one instinctively wants to complete the ...
75 views

### number of generators of MASA

Let $\mathcal{H}$ be an infinite-dimensional Hilbert Space. Do the maximal abelian self-adjoint subalgebras of $\mathcal{B}(\mathcal{H})$ always have infinitely many generators as an algebra ? (The ...
49 views

### Are all the square roots of non-square numbers surds? [duplicate]

Quite self explanatory really, basically, are $\sqrt5 ,\sqrt3$ and $\sqrt7$ and surds? (So basically, every square root of any non-square number)
28 views

### Order embedding and graph embedding of Hasse digraphs

If there is an order embedding from order A to order B, is there a graph embedding between their Hasse digraphs? What if we replace 'embedding' by 'order preserving' and 'homomorphism' etc?
67 views

### An inner product on a space of linear maps

Let $V$ and $H$ be two complex Hilbert spaces. We suppose $V$ to be finite-dimensional. I'd like to understand the structure of Hilbert space on the space of linear mappings $\mathrm{Hom}(V,H)$. ...
222 views

### Principle of recursion for inductively defined relations

If we consider a relation $R$ and then its symmetric-reflexive-transitive closure --say $R^*$--, is there a recursion principle associated with $R^*$? It seems to me that such unique function is not ...
168 views

### How fast is the water draining out after 5 min?

The volume $V$, in liters, of water in a water tank after $t$ min it starts draining, is given by $$V(t)=260(60−t)^2$$ How fast is the water draining out after 5 min? Do I calculate the ...
73 views

### conjugate prime ideals of integral extensions and relevance of the characteristic of the ground field

This question refers to the proof of theorem 9.3, p. 66 in Matsumura's Commutative Ring Theory: "if $A$ is an integrally closed domain, $K$ its field of fractions and $L/K$ a normal field extension, ...
63 views

### What are co-products for directed graphs?

I define a digraph as a set $V$ (vertexes) and a relation $E$ (edges) on $V$. Morphisms of digraph are functions which preserve $E$. So we have a category. What are co-products in this category? (I ...