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Introduction to Linear Algebra with Mathematica

Glossary

Preface

First Order PDEs

WE consider complex-valued functions depending on two variables: one is a spacial variable and another one corresponds time, so it is considered nonnegative: u, v : ℝd × ℝ≥0 → ℂ, so
\[ u(x,t) = \begin{pmatrix} u_1 (x, t) \\ u_2 (x,t) \\ \vdots \\ u_d (x,t) \end{pmatrix} , \qquad x \in \mathbb{R}^d , \quad t \ge 0 . \]
Assuming that these functions are 2π-periodic, we introduce the inner product
\[ \langle u , v \rangle = \int _{0}^{2\pi}\cdots \int _{0}^{2\pi}\int _{0}^{2\pi}\left\langle u\left(\cdot ,t\right),v\left(\cdot ,t\right)\right\rangle {\text d}x_1 {\text d}x_2\cdots {\text d}x_d , \]
where ⟨ u , v ⟩ = u*1 v1 + u*2 v2 + ⋯ + u*d vd is the dot product in ℂd. This inner product induces a norm ∥ · ∥ = √⟨·,·⟩. The inner product generates the norm in the space 𝔏²[0, 2π], the set of all Lebesgue integrable functions with finite norm.

 

Orthoniormal basis in 𝔏²[0, 2π]

We will first show that
\[ \left\{\frac{1}{(2\pi)^{d/2}}\:e^{i\langle w,x \rangle}e_j \: \bigg| \: w \in \mathbb{Z}^d, 1 \leq j \leq m\right\} \]
with x ∈ ℝd, is an orthonormal set with respect to the Hilbert space 𝔏²[0, 2π].

Let x, u, v ∈ ℝd

 

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