This series of study notes is aimed for post-graduate (M.A/M.Sc.) students of Indian & international universities. The study of functional analysis can be started after basic topology and set theory courses. In this introductory article we will start with some elementary yet important definitions and notations from analysis. We will finish this article with the definition of Norm & Normed Linear spaces derivedfrom the notions of linear spaces. An elementary treatise of examples and their completeness of Normed (linear) Spaces will be done in upcoming articles. Using this we shall define complete normed spaces or Banach Spaces. After basic ideas are attained about Normed & Banach spaces, there will be rigorous discussion on their subspaces, quotient spaces. A relevant analysis of properties like joint continuity of addition and vector multiplication, equivalent norms, compactness, boundedness etc. will be done at last.

Topic suggestions, improvement tips and further queries are welcome through comment form or via e-mail.

CHECK OUT THE LIST OF NOTATIONS USED IN MATH ARTICLES ON MY MAGAZINE

Linear Space or Vector Space over a field K

Definition: The linear space over a field K is a non-empty set along with a function $+ : X \times X \to X$ called linear/vector addition  (or just, ‘addition‘) and another function $ \cdot : K \times X \to X$ called scalar multiplication, such that for all elements $x, y, z, \ldots$ in $X$ and $1, k, l, \ldots$ in $K$ :

$x+y = y+x$
$x+(y+z) = (x+y)+z$
there exists $0 \in X$ such that $x+0=x$
there exists $-x \in X$ such that $x+(-x)=0$
$k \cdot (x+y)=k \cdot x+k \cdot y$
$(k+l) \cdot x = k \cdot x + l \cdot x$
$(kl) \cdot x = k \cdot (l \cdot x)$
$1 \cdot x =x$

Set-Set, Set Element Summation & Products

Let small Roman letters like $x, y, a, b, \ldots$ represent the elements & capital letters like $A, B, X, Y, \ldots$ represent sets, then

  • $x+F = \{ x+y : y \in F \} $
  • $ E+F= \{ x+y: x \in E, \, y \in F \}$
  • $kE= \{ kx : x \in E \}$
  • $E \times F = \{ (x,y) : x \in E, \, y \in F \}$

Convex Subset of a linear space

A subset E of a linear space X over field K is said to be convex if $rx+(1-r)y \in E$ when $x,y \in E$ and $0<r<1$

Convex Hull of a subset of linear space

For $E \subset X$, the smallest convex subset of linear space X containing $E$ is called the convex hull of $E$, denoted by co(E).

co(E)= $ \{ \displaystyle{\sum_{i=1}^n} r_i x_i : x_i \in E; r_i \ge 0; \displaystyle{\sum_{i=1}^n} r_i=1\}$

Subspace of a linear space

A non-empty subset $Y$ of linear space $X$ over K is said to be a subspace of $X$ over K if $kx+ly \in Y$, whenever $x,y \in Y$ & $k,l \in K$.

Span of a subset of linear space

For a non-empty subset E of linear space X over K, the smallest subspace of X containing E is span(E) defined as

span(E) = $ \{ \displaystyle{\sum_{i=1}^n} k_i x_i : x_i \in E; k_i \in K \}$

This set is called the span of E.

REMARK: When span(E)=X , then we say that E spans X. Also, if span(E)=X and is a linearly independent set, E is called the Hamel Basis (or basis) of linear space X.

Linear Map

Let X and Y be two linear spaces over K. A linear map from X to Y is a function F : X →Y such that $F(k_1 x_1 + k_2 x_2) = k_1 F(x_1)+ k_2 F(x_2)$ for all $x_1, x_2 \in X, k_1, k_2 \in K$

The subspace $$ R(F):= \{ y \in Y : F(x)=y \, \mathbf{for \, some} \, x \in X \} $$ of Y is called the range space$ of F. While, the subspace $$ Z(F) := \{ x \in X : F(x) = 0 \}$$ of X is called the zero space$ of F.

REMARK:

  • Whenever Z(F)=X, we write F=0.
  • dim X= dim R + dim Z

 

Norm

Let X be a linear space over the field K of real or complex numbers. A norm on X is the function $|| \, || : X \to R$ such that for all $x, y \in X$ and $k \in K$,

  • $||x|| \ge 0$ with $||x||=0$ if and only if $x=0$
  • $||x+y|| \le ||x||+||y||$
  • $||kx|| = |k| ||x||$ where $|k|$ is the modulus of $k$.

A normed space X is a linear space with a norm ||  || on it.

Examples of Normed Space

A descriptive analysis of following normed spaces will be done in next article:

  • Spaces $\mathbb{R}^n$ and $\mathbb{C}^n$
  • Sequence spaces $l^p, l^\infty, c, c_0, c_{00}$ where $1\le p <\infty$
  • p-integrable function spaces $L^p, L^\infty$ where $1\le p <\infty$

 

Total
0
Shares


Feel free to ask questions, send feedback and even point out mistakes. Great conversations start with just a single word. How to write better comments?
Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

You May Also Like

The Mystery of the Missing Money – One Rupee

Puzzle Two women were selling marbles in the market place — one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two lots of marbles and handing them over to a friend asked her to sell then…

Four way valid expression

People really like to twist the numbers and digits bringing fun into life. For example, someone asks, “how much is two and two?” : the answer should be four according to basic (decimal based) arithmetic. But the same  with base three (in ternary number system) equals to 11. Two and Two also equals to Twenty Two. Similarly there are many ways you can…

When nothing is everything in Set theory

In an earlier post, I discussed the basic and most important aspects of Set theory, Functions and Real Number System. In the same, there was a significant discussion about the union and intersection of sets. Restating the facts again, given a collection $ \mathcal{A}$ of sets, the union of the elements of $ \mathcal{A}$ is defined by $ \displaystyle{\bigcup_{A \in…

Best Time Saving Mathematics Formulas & Theorems

Formulas are the most important part of mathematics and as we all know one is the backbone of the latter. Considering there are thousands of mathematical formulas to help people develop analytical approach and solve problems easily — there are some that go beyond. Some formulas aren’t just timesaving but those also do wonders. In this article I have collected…

How Many Fishes in One Year? [A Puzzle in Making]

This is a puzzle which I told to my classmates during a talk, a few days before. I did not represent it as a puzzle, but a talk suggesting the importance of Math in general life. This is partially solved for me and I hope you will run your brain-horse to help me solve it completely. If you didn’t notice,…

The ‘new’ largest known Prime Number

Great Internet Mersenne Prime Search (GIMPS) group has reported an all new Mersenne Prime Number (a prime number of type $2^P-1$) which is, now officially the largest prime number ever discovered. This number is valued to a whopping $2^{74207281}-1$ and contains 22,338,618 digits. It is quoted as M747207281 and is almost 5 million digits longer than the previous record holding prime number…