Question

2. द्वैत-समष्टि को उदाहरण सहित परिभाषित कीजिए। Define dual space with example.
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Answer 1

Answer:

In mathematics, any vector space {\displaystyle V}V has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on {\displaystyle V}V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.

Answer 2

Definition:

Suppose V is a vector space over a field F. Then the collection of linear maps φ:V→F is called the Dual Space of V.

Example:

Suppose V is a finite-dimensional vector space with the basis {e₁,e₂,...,eₙ}. So, we take {e¹,e²,...,eⁿ} as the dual basis of linear functionals on V, which are defined by the relation

$\mathbf{e}^i(c^1 \mathbf{e}_1+\cdots+c^n\mathbf{e}_n) = c^i, \quad i=1,\ldots,n$ for any choice of coefficients $c_i \in F.$
where, $\mathbf{e}^i(\mathbf{e}_j) = \delta^{i}_{j}$

where, where ${\displaystyle \delta _{j}^{i}}$ is the Kronecker delta symbol i.e.

${\displaystyle \delta _{j}^{i}}=0 \mbox{ if }i\neq j\\\mbox{ }\mbox{ }\mbox{ } =1 \mbox{ if }i= j$

Hence, V*=Span{e¹,e²,...,eⁿ} is the Dual Space of V.

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