MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH4313 Functional Analysis信息论课程的代写代考和辅导服务！
这是悉尼大学 泛函分析课程的代写成功案例。
MATH4313课程简介
Functional analysis is one of the major areas of modern mathematics. It can be thought of as an infinite-dimensional generalisation of linear algebra and involves the study of various properties of linear continuous transformations on normed infinite-dimensional spaces. Functional analysis plays a fundamental role in the theory of differential equations, particularly partial differential equations, representation theory, and probability. In this unit you will cover topics that include normed vector spaces, completions and Banach spaces; linear operators and operator norms; Hilbert spaces and the Stone-Weierstrass theorem; uniform boundedness and the open mapping theorem; dual spaces and the Hahn-Banach theorem; and spectral theory of compact self-adjoint operators. A thorough mechanistic grounding in these topics will lead to the development of your compositional skills in the formulation of solutions to multifaceted problems. By completing this unit you will become proficient in using a set of standard tools that are foundational in modern mathematics and will be equipped to proceed to research projects in PDEs, applied dynamics, representation theory, probability, and ergodic theory.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. demonstrate a coherent and advanced understanding of the key concepts of geometry of normed spaces, Hilbert Space Theory, Abstract Fourier Analysis, Hahn-Banach Theory and Spectral Theory, and how they provide a unified approach to infinite-dimensional linear problems in mathematics
- LO2. apply the fundamental ideas and results in functional analysis to solve given problems
- LO3. distinguish and compare the properties of different types of linear operators, analysing their spectra and deriving their main properties
- LO4. formulate analytic problems in functional-analytic terms and determine the appropriate framework to solve them
- LO5. communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work
- LO6. devise computational solutions to complex problems in functional analysis
- LO7. compose correct proofs of unfamiliar general results in functional analysis.
MATH4313 Functional Analysis HELP（EXAM HELP， ONLINE TUTOR）
Suppose $f: \mathbf{R} \rightarrow \mathbf{R}$ is continuous and 1-periodic, namely $f(x+1)=f(x)$ for all $x \in \mathbf{R}$. Prove that if $\gamma \in \mathbf{R}$ is irrational, then
$$
\lim {n \rightarrow \infty} \sum{j=1}^n f(j \gamma)=\int_0^1 f(x) d x .
$$
(This is a special case of the Birkhoff ergodic theorem.)
Suppose that $M$ is a closed subspace of a Hilbert space $H$. Fix $x \in H$ and define $x+M \stackrel{\text { def }}{=}{x+y$ : $y \in M}$.
a. Prove that $x+M$ is a closed convex subset of $H$.
b. Let $Q x$ be the (unique) point of $x+M$ with smallest norm and let $P x=x-Q x$. ( $P$ is called the orthogonal projection of $x$ onto $M$.) Prove that $P$ and $Q$ are surjective from $H$ to $M$ and $M^{\perp}$, respectively.
c. Prove that $P$ and $Q$ are linear mappings.
d. Prove that if $y \in M$ then $P y=y$ and $Q y=0$.
e. Prove that if $z \in M^{\perp}$ then $P z=0$ and $Q z=z$.
f. Prove that $|w|^2=|P w|^2+|Q w|^2$ for any $w \in H$.
Suppose $\left{e_n\right}$ is a countable orthonormal basis for a Hilbert space $H$ and $\left{f_n\right}$ is a countable orthonormal set such that
$$
\sum_n\left|e_n-f_n\right|^2<1 .
$$
Prove that $\left{f_n\right}$ is a basis.
Suppose that $\left{e_n\right}$ and $\left{f_n\right}$ are countable orthonormal bases for a Hilbert space $H$. Define a linear transformation $T: H \rightarrow H$ by
$$
T\left(\sum_n c_n e_n\right)=\sum_n c_n f_n .
$$
a. Prove that $T$ is continuous and compute the operator norm of $T$.
b. Prove that $\langle T x, T y\rangle=\langle x, y\rangle$ for all $x, y \in H$.
Let $H, G$ be Hilbert spaces. Say that a linear function $T: H \rightarrow G$ is an isometry if $\langle T x, T y\rangle_G=\langle x, y\rangle_H$ for every $x, y \in H$.
a. Prove that an isometry $T$ is continuous and compute the operator norm of $T$.
b. Prove that if $T$ is an isometry then it is injective.
c. Must an isometry be surjective? Supply a proof or a counterexample.
d. Suppose $T_1$ and $T_2$ are isometries. Must $T_1+T_2$ be an isometry?
Let $X, Y$ be Hilbert spaces. Suppose that $T: X \rightarrow Y$ is a bounded linear function. Prove that there exists a unique bounded linear function $T^: Y \rightarrow X$ satisfying $$ (\forall x \in X)(\forall y \in Y)\langle T x, y\rangle_Y=\left\langle x, T^ y\right\rangle_X .
$$
(Such $T^*$ is called the adjoint of $T$.)
MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH4313 Functional Analysis信息论课程的代写代考和辅导服务！
