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.
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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 E 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 $1\le p <\infty$
