Clarifying Notation $H : \mathbb{R}^n \to \mathbb{R}, (h_1, \dots, h_n) \mapsto \frac{1}{2} \sum_{i, j = 1}^n b_{ij} h_i h_j$

Question

My textbook presents the following lemma:

Lemma $1$ If $B = [b_{ij}]$ is an $n \times n$ real matrix, and if the associated quadratic function

$$H : \mathbb{R}^n \to \mathbb{R}, (h_1, \dots, h_n) \mapsto \dfrac{1}{2} \sum_{i, j = 1}^n b_{ij} h_i h_j$$

is positive-definite, then there is a constant $M > 0$ such that for all $\mathbf{h} \in \mathbb{R}^n$;

$$H(\mathbf{h}) \ge M ||\mathbf{h}||^2$$

I'm unfamiliar with this function notation:

$$H : \mathbb{R}^n \to \mathbb{R}, (h_1, \dots, h_n) \mapsto \dfrac{1}{2} \sum_{i, j = 1}^n b_{ij} h_i h_j$$

1. I am not sure what it means to join the two parts $H : \mathbb{R}^n \to \mathbb{R}$ and $(h_1, \dots, h_n) \mapsto \dfrac{1}{2} \sum_{i, j = 1}^n b_{ij} h_i h_j$, separated with a comma?

2. The textbook always used the arrow $\to$, such as in $H : \mathbb{R}^n \to \mathbb{R}$, but I'm unfamiliar with what the addition of the arrow $\mapsto$ means here? And what is the difference between $\mapsto$ and $\to$?

3. After clarifying the above two points, putting everything in a cohesive picture, what is meant by $$H : \mathbb{R}^n \to \mathbb{R}, (h_1, \dots, h_n) \mapsto \dfrac{1}{2} \sum_{i, j = 1}^n b_{ij} h_i h_j?$$

I would greatly appreciate it if people could please take the time to clarify this.

Answer 1

The two parts, separated by a comma, simply say two things:

1. The function $H$ maps from $\mathbb R^n$ to $\mathbb R$. In other words, the domain of $H$ is $\mathbb R^n$ and the codomain of $H$ is $\mathbb R$.
2. For an input $(h_1,\dots, h_n)$, the output of the function is $\dfrac{1}{2} \sum_{i, j = 1}^n b_{ij} h_i h_j$

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Remember, a function is defined by three things: Its domain, codomain and its... let's call it "action".

The fact that the domain and codomain of a function $f$ as $X$ and $Y$ is usually written like so:

$$f: X\to Y$$

while the "action" of the function is written in one of two ways. One is to simply say $$f(x)=\dots,$$ the other is to say $$f:x\mapsto \dots$$ In either way, we write the same thing in the dots.

Answer 2

The first bit defines the domain and the codomain of the function $$H:\mathbb R^n\rightarrow \mathbb R$$ just says that the function $H$ takes as entries elements of $\mathbb R^n$, which are vectors like $(h_1, h_2, \dotso, h_n)$, and maps them into elements of $\mathbb R$ which are just numbers. But how does this function map such elements into numbers? You have now to define the explicit form of the function which is the second bit $$(h_1,h_2,\dotso, h_n)\mapsto \frac{1}{2}\sum_{i,j=1}^n b_{ij}h_ih_j$$ which is the same as saying $$H(h_1,h_2,\dotso, h_n)= \frac{1}{2}\sum_{i,j=1}^n b_{ij}h_ih_j$$ So for short $\mapsto$ indicates how the function maps certain elements to others, and the symbol $\rightarrow$ indicates from what domain into what codomain the function operates.