Tagged Questions
1
vote
1answer
27 views
Terminology of functions in $L_2$
I am reading a text that states
... any function in $L_2(0,\pi)$ has a Fourier sine series that
converges to it in $L_2(0,\pi)$ ...
Unfortunately no definition of $L_2$ is given. What does ...
1
vote
1answer
23 views
What is the name for functions with this property similar to concavity?
What's the name of the class of functions satisfying the following property:
Given some convex space $D\subseteq \mathbb{R}^k$, a function $f:D \to \mathbb{R}$ is called $\mathbf{(?)}$ if for all ...
2
votes
1answer
51 views
What does this phrase about the weak topology of bounded operators mean?
Can somenone remind me of the meaning of the following statement:
the family of operator valued functions $A(\omega)$ converges to $A(\omega ')$ in the weak topology of bounded operators from ...
1
vote
2answers
30 views
I'm having trouble with math terminology, when is it appropriate to say “mapping” or “maps to”?
For example, in a homework assignment I need to prove that a metric space $M$ is isometric to itself. If I say that $M$ maps to $M$ is that the same as saying that theres a function that takes ...
2
votes
1answer
57 views
What is meant by the word ‘control’ in the context of analysis?
Wikipedia’s article on Dini’s Theorem states:
This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by ...
1
vote
1answer
242 views
Defintion of Upper & Lower Riemann Sum
I recently came across the terms: 'upper Riemann sum' and 'lower riemann sum'. Are they represent the same things as of 'upper sum' and 'lower sum' defined as follows.
0
votes
1answer
65 views
What does $f$ is measurable as a function $f^{-1}(\mathbb{R})\to\mathbb{R}$ mean?
I saw a question where we have $\overline{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\}$
and $(X,S)$ is a measurable space, $f:\, X\to\overline{\mathbb{R}}$
In one part I was told to assume that $f$ is ...
0
votes
1answer
35 views
What does it mean for a set to be “nested”?
What does it mean for a set to be "nested"?
and can you please show an example of that is and one that isn't
1
vote
1answer
161 views
Name for the real numbers between $0$ and $1$
I see this class of numbers all the time, so I was wondering if there was a special name for it.
How to refer to a number $n$ in $\Bbb R$, such that $0<n<1$?
3
votes
0answers
92 views
Property similar to subadditivity
A function is called subadditive such
that $f(x+y)\le f(x)+f(y)$ holds for any $x$, $y$ in the domain of $f$. (Let us say that, for example,
the domain is some subset of $\mathbb R$ closed under ...
1
vote
1answer
199 views
Is there a different name for strongly Darboux functions?
A function $f\colon\mathbb R\to\mathbb R$ is called Darboux function, function with Darboux property or function with intermediate value property for for any $a<b$ and any $z$ between $f(a)$ and ...
3
votes
1answer
1k views
Cluster point of a function at a point
This post is quite long, since I wanted to include the necessary context. (Maybe I've put there too much.) Maybe you might prefer to look at the questions at the end of this post, first.
...
1
vote
3answers
735 views
why do we use 'non-increasing' instead of decreasing?
In english based math language it seems that
non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing)
decreasing $\Longleftrightarrow$ strict less ( strict decreasing)
...
2
votes
0answers
236 views
Are derivative and differential the same thing?
(Sorry for bad English.)
Let $f:\mathbb R^n\to\mathbb R^m$, $x\in\mathbb R^n$. What is a drivative $f'(x)$? It is the linear map $f'(x):\mathbb R^n\to\mathbb R^m$, $h\mapsto ...
3
votes
3answers
228 views
Is there a name encompassing both limit inferior and limit superior
Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?)
The only thing I was able to find was that some ...
8
votes
1answer
173 views
What kind of completeness is the completeness of $\mathbb{R}$?
As opposed to the algebraic completeness of $\mathbb{Q}$, which yields the algebraic numbers, we can say that $\mathbb{R}$ is complete in the sense that every non-empty subset of $\mathbb{R}$ bounded ...
14
votes
8answers
3k views
Why does “convex function” mean “concave *up*”?
A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$,
$$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$
Equivalently, a line ...
