• Title: Dual Spaces

  • Series: Functional Analysis

  • Chapter: Core Results in Functional Analysis

  • YouTube-Title: Functional Analysis 22 | Dual Spaces

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  • Subtitle on GitHub: fa22_sub_eng.srt missing

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  • Timestamps

    00:00 Introduction

    00:51 Definition of dual space $X^\prime$

    01:45 Correction: Riesz says $X$ and $X^\prime$ are connected by an conjugate-linear bijection that is also isometric

    02:43 Proposition: $X^\prime$ is always complete.

    02:55 Proof that dual space is Banach space.

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  • Quiz Content

    Q1: Let $X$ be a normed space over the field $\mathbb{F}$. What is the (topological) dual space of $X$, denoted by $X^\prime$?

    A1: ${ \ell: X \rightarrow \mathbb{F} $ $~\mid \ell \text{ linear + bounded} }$

    A2: ${ \ell: X \rightarrow \mathbb{F} $ $~\mid \ell \text{ linear } }$

    A3: ${ \ell: X \rightarrow \mathbb{F} $ $~\mid \ell \text{ monotonic } }$

    A4: ${ \ell: X \rightarrow \mathbb{F} $ $~ \mid \ell \text{ linear + unbounded } }$

    A5: ${ \ell: X \rightarrow \mathbb{F} $ $~\mid \ell \text{ non-negative } }$

    Q2: Let $X$ be a normed space. Is the dual space $X^\prime$ a Banach space?

    A1: Yes, it is.

    A2: No, it never is a Banach space.

    A3: It’s only a Banach space in the case that $X$ is also a Banach space.

    A4: It’s only a Banach space in the case that $X$ is not complete.

    Q3: Let $X$ be a finite dimensional Banach space. What is not correct for the dual space $X^\prime$?

    A1: $X^\prime$ is a finite dimensional Banach space.

    A2: $X^\prime$ is infinite dimensional.

    A3: $X^\prime$ has the same dimension as $X$.

    A4: $X^\prime$ is a normed space with the operator norm.

  • Last update: 2025-09

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