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Normed and Banach spaces

Vector spaces with a length; completeness; the classical sequence and function spaces.

A norm measures length; a Banach space is a normed space with no missing limits. ℓ^p, L^p and the continuous functions with the sup norm are the standard examples. Picture it: the unit balls for different norms, a diamond, a circle, a square. Think it: completeness is what lets you solve equations by successive approximation (the contraction mapping theorem).

Àwọn ààtò àwọn ìṣàmúlò-ètò · sqrt(3^2 + 4^2)

Arithmetic · Arithmetic

Evaluate sqrt(3^2 + 4^2)

\sqrt{3^{2} + 4^{2}}

Àwọn ìṣàmúlò-ètò

\sqrt{3^{2} + 4^{2}} = \sqrt{9 + 4^{2}}
Power: 3^2 = 9.
\sqrt{9 + 4^{2}} = \sqrt{9 + 16}
Power: 4^2 = 16.
\sqrt{9 + 16} = \sqrt{25}
Add: 9 + 16 = 25.
\sqrt{25} = 5
Simplify.

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Bii o ṣe le: Normed and Banach spaces

Power: 3^2 = 9.
Power: 4^2 = 16.
Add: 9 + 16 = 25.
Simplify.

Àwọn Àtòjọ-ẹ̀yàn

What is a Hilbert space?

A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.

Diẹ̀ nínú Functional Analysis

Hilbert spaces and the spectral theorem