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 $M$ is a subspace of a Hilbert space $H$.
a. Prove that if $M$ is closed, then $\left(M^{\perp}\right)^{\perp}=M$.
b. Find a counterexample where $M$ is not closed and $\left(M^{\perp}\right)^{\perp} \neq M$.
Prove that if $H$ is infinite-dimensional, namely there are linearly independent subsets $\left{x_1, \ldots, x_n\right}$ for all $n=1,2, \ldots$, then the closed unit ball in $H$ is not compact.
Suppose $\left{a_n: n=1,2, \ldots\right}$ is a sequence of real numbers such that
$$
\sum_{n=1}^{\infty} a_n b_n<\infty
$$
whenever $\sum_{n=1}^{\infty} b_n^2<\infty$. Prove that $\sum_{n=1}^{\infty} a_n^2<\infty$.
Say that $x_n \rightarrow x$ weakly in a Hilbert space $H$ if $\left\langle x_n, y\right\rangle \rightarrow\langle x, y\rangle$ for every $y \in H$. Prove that if $x_n$ is a sequence in $H$ bounded by $\sup n\left|x_n\right| \leq 1$, then there is a subsequence $n_j$ and an element $x \in H$ with $|x| \leq 1$ such that $x{n_j} \rightarrow x$ weakly as $j \rightarrow \infty$.
If $A$ is a Lebesgue measurable subset of $[0,2 \pi)$, prove that
$$
\lim _{n \rightarrow \infty} \int_A e^{i n x} d x=0 .
$$
(This is a special case of the Riemann-Lebesgue lemma.)
Suppose that $\mu, \nu$ are finite measures on a measurable space $(X, \mathcal{A})$, with $\nu \ll \mu$. Assume that $\nu(A) \leq \mu(A)$ for all measurable $A$.
For real-valued $f \in L^2(X, \mu)$, define $L(f) \stackrel{\text { def }}{=} \int_X f d \mu$.
a. Show that $L$ is a bounded linear functional on $L^2(X, \mu)$.
b. Show that there exists a real-valued measurable function $g \in L^2(X, \mu)$ such that $L(f)=\int_X f g d \mu$ for all $f \in L^2(X, \mu)$. (Hint: use theorem 19.10 on textbook p.188.)
c. Show that $g$ from part b is the Radon-Nikodym derivative $d \nu / d \mu$.
Note: by exercise 13.6 on textbook p.105, the assumption $\nu(A) \leq \mu(A)$ imposes no restriction as one may replace $\mu$ with $\mu+\nu$. Hence, this is an alternate proof of the Radon-Nikodym theorem as a corollary of theorem 19.10 .
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！
