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1. Foreword 2 2. Preliminaries 3 2.1. Symmetric powers 3 2.2. Concatenation and splitting 5 2.3. Measurability 6 2.4. Lipschitz extensions 6 2.5. Differentiability 13 3. Embeddings 21 3.1. Whitney bi-Hölder embedding — The case Y “ l2 pKq 21 3.2. Splitting in case Y “ R 21 3.3. Almgren-White locally isometric embedding — The case Y “ l2 pRq 22 3.4. Lipeomorphic embedding into Lipy0pY q ̊ 27 4. Sobolev classes 28 4.1. Definition of LppX,QQpY qq 28 4.2. Analog of the Fréchet-Kolmogorov compactness Theorem 29 4.3. Definition of W 1 p pU ;QQpY qq 31 4.4. The p energy 34 4.5. Extension 39 4.6. Poincaré inequality and approximate differentiability almost everywhere 41 4.7. Trace 46 4.8. Analog of the Rellich compactness Theorem 49 4.9. Existence Theorem 50 References 50

Existence of p harmonic multiple valued maps into a separable Hilbert space